Theorem sbcssgVD 39936

Description: Virtual deduction proof of sbcssg 4307. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sbcssg 4307 is sbcssgVD 39936 without virtual deductions and was automatically derived from sbcssgVD 39936.

1::	⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
2:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)   )
3:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐷 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)   )
4:2,3:	⊢ (   𝐴 ∈ 𝐵   ▶   (([𝐴 / 𝑥]𝑦 ∈ 𝐶 → [𝐴 / 𝑥]𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷 ))   )
5:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 → [𝐴 / 𝑥]𝑦 ∈ 𝐷))   )
6:4,5:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   )
7:6:	⊢ (   𝐴 ∈ 𝐵   ▶   ∀𝑦([𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   )
8:7:	⊢ (   𝐴 ∈ 𝐵   ▶   (∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷) )   )
9:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷))   )
10:8,9:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷) )   )
11::	⊢ (𝐶 ⊆ 𝐷 ↔ ∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷))
110:11:	⊢ ∀𝑥(𝐶 ⊆ 𝐷 ↔ ∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷))
12:1,110:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ [𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷))   )
13:10,12:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   )
14::	⊢ (⦋𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷 ↔ ∀ 𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))
15:13,14:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷)   )
qed:15:	⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ ⦋ 𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷))

(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sbcssgVD	⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷))

Proof of Theorem sbcssgVD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		idn1 39617	. . . . . . . . . 10 ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
2		sbcel2 4215	. . . . . . . . . . 11 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)
3	2	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))
4	1, 3	e1a 39679	. . . . . . . . 9 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)   )
5		sbcel2 4215	. . . . . . . . . . 11 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐷 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)
6	5	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝑦 ∈ 𝐷 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))
7	1, 6	e1a 39679	. . . . . . . . 9 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐷 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)   )
8		imbi12 338	. . . . . . . . 9 ⊢ (([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶) → (([𝐴 / 𝑥]𝑦 ∈ 𝐷 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷) → (([𝐴 / 𝑥]𝑦 ∈ 𝐶 → [𝐴 / 𝑥]𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))))
9	4, 7, 8	e11 39740	. . . . . . . 8 ⊢ (   𝐴 ∈ 𝐵   ▶   (([𝐴 / 𝑥]𝑦 ∈ 𝐶 → [𝐴 / 𝑥]𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   )
10		sbcimg 3704	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 → [𝐴 / 𝑥]𝑦 ∈ 𝐷)))
11	1, 10	e1a 39679	. . . . . . . 8 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 → [𝐴 / 𝑥]𝑦 ∈ 𝐷))   )
12		bibi1 343	. . . . . . . . 9 ⊢ (([𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 → [𝐴 / 𝑥]𝑦 ∈ 𝐷)) → (([𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) ↔ (([𝐴 / 𝑥]𝑦 ∈ 𝐶 → [𝐴 / 𝑥]𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))))
13	12	biimprcd 242	. . . . . . . 8 ⊢ ((([𝐴 / 𝑥]𝑦 ∈ 𝐶 → [𝐴 / 𝑥]𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) → (([𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 → [𝐴 / 𝑥]𝑦 ∈ 𝐷)) → ([𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))))
14	9, 11, 13	e11 39740	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   )
15	14	gen11 39668	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ▶   ∀𝑦([𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   )
16		albi 1917	. . . . . 6 ⊢ (∀𝑦([𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) → (∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)))
17	15, 16	e1a 39679	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ▶   (∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   )
18		sbcal 3712	. . . . . . 7 ⊢ ([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷))
19	18	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷)))
20	1, 19	e1a 39679	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷))   )
21		bibi1 343	. . . . . 6 ⊢ (([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷)) → (([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) ↔ (∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))))
22	21	biimprcd 242	. . . . 5 ⊢ ((∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) → (([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷)) → ([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))))
23	17, 20, 22	e11 39740	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   )
24		dfss2 3815	. . . . . 6 ⊢ (𝐶 ⊆ 𝐷 ↔ ∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷))
25	24	ax-gen 1894	. . . . 5 ⊢ ∀𝑥(𝐶 ⊆ 𝐷 ↔ ∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷))
26		sbcbi 39582	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝐶 ⊆ 𝐷 ↔ ∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷)) → ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ [𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷))))
27	1, 25, 26	e10 39746	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ [𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷))   )
28		bibi1 343	. . . . 5 ⊢ (([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ [𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷)) → (([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) ↔ ([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))))
29	28	biimprcd 242	. . . 4 ⊢ (([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷) ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) → (([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ [𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷)) → ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))))
30	23, 27, 29	e11 39740	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   )
31		dfss2 3815	. . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷 ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))
32		biantr 840	. . . 4 ⊢ ((([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) ∧ (⦋𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷 ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))) → ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷))
33	32	ex 403	. . 3 ⊢ (([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) → ((⦋𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷 ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) → ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷)))
34	30, 31, 33	e10 39746	. 2 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷)   )
35	34	in1 39614	1 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 ⊆ 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐶 ⊆ ⦋𝐴 / 𝑥⦌𝐷))

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1654   ∈ wcel 2164  [wsbc 3662  ⦋csb 3757   ⊆ wss 3798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-in 3805  df-ss 3812  df-nul 4147  df-vd1 39613

This theorem is referenced by: (None)