Theorem sbcim2gVD 45816

Description: Distribution of class substitution over a left-nested implication. Similar to sbcimg 3787. 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. sbcim2g 45480 is sbcim2gVD 45816 without virtual deductions and was automatically derived from sbcim2gVD 45816.

1::	⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
2::	⊢ (   𝐴 ∈ 𝐵   ,   [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))   ▶   [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))   )
3:1,2:	⊢ (   𝐴 ∈ 𝐵   ,   [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))   ▶   ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))   )
4:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜓 → 𝜒) ↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))   )
5:3,4:	⊢ (   𝐴 ∈ 𝐵   ,   [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))   ▶   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))   )
6:5:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))   )
7::	⊢ (   𝐴 ∈ 𝐵   ,   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))   ▶   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))   )
8:4,7:	⊢ (   𝐴 ∈ 𝐵   ,   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))   ▶   ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))   )
9:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒)))   )
10:8,9:	⊢ (   𝐴 ∈ 𝐵   ,   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))   ▶   [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))   )
11:10:	⊢ (   𝐴 ∈ 𝐵   ▶   (([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)))   )
12:6,11:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))   )
qed:12:	⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))))

(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sbcim2gVD	⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))))

Proof of Theorem sbcim2gVD

Step	Hyp	Ref	Expression
1		idn1 45516	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
2		idn2 45555	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ,   [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))   ▶   [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))   )
3		sbcimg 3787	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))))
4	3	biimpd 232	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))))
5	1, 2, 4	e12 45665	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ,   [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))   ▶   ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))   )
6		sbcimg 3787	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜓 → 𝜒) ↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))
7	1, 6	e1a 45569	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜓 → 𝜒) ↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))   )
8		imbi2 351	. . . . . 6 ⊢ (([𝐴 / 𝑥](𝜓 → 𝜒) ↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → (([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))))
9	8	biimpcd 252	. . . . 5 ⊢ (([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒)) → (([𝐴 / 𝑥](𝜓 → 𝜒) ↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))))
10	5, 7, 9	e21 45671	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ,   [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))   ▶   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))   )
11	10	in2 45547	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))   )
12		idn2 45555	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ,   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))   ▶   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))   )
13		biimpr 223	. . . . . . 7 ⊢ (([𝐴 / 𝑥](𝜓 → 𝜒) ↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → (([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒) → [𝐴 / 𝑥](𝜓 → 𝜒)))
14	13	imim2d 58	. . . . . 6 ⊢ (([𝐴 / 𝑥](𝜓 → 𝜒) ↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → (([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))))
15	7, 12, 14	e12 45665	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ,   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))   ▶   ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))   )
16	1, 3	e1a 45569	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒)))   )
17		biimpr 223	. . . . . 6 ⊢ (([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))) → (([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒)) → [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))))
18	17	com12 33	. . . . 5 ⊢ (([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒)) → (([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))) → [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))))
19	15, 16, 18	e21 45671	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ,   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))   ▶   [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))   )
20	19	in2 45547	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   (([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)))   )
21		impbi 211	. . 3 ⊢ (([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))) → ((([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))) → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))))
22	11, 20, 21	e11 45630	. 2 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))   )
23	22	in1 45513	1 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2145  [wsbc 3739

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-sbc 3740  df-vd1 45512  df-vd2 45520

This theorem is used by: (None)