Theorem csbeq2gVD 45465

Description: Virtual deduction proof of csbeq2 3860. 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. csbeq2 3860 is csbeq2gVD 45465 without virtual deductions and was automatically derived from csbeq2gVD 45465.

1::	⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   )
2:1:	⊢ (   𝐴 ∈ 𝑉   ▶   (∀𝑥𝐵 = 𝐶 → [𝐴 / 𝑥] 𝐵 = 𝐶)   )
3:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)   )
4:2,3:	⊢ (   𝐴 ∈ 𝑉   ▶   (∀𝑥𝐵 = 𝐶 → ⦋𝐴 / 𝑥 ⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)   )
qed:4:	⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌ 𝐵 = ⦋𝐴 / 𝑥⦌𝐶))

(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
csbeq2gVD	⊢ (𝐴 ∈ 𝑉 → (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶))

Proof of Theorem csbeq2gVD

Step	Hyp	Ref	Expression
1		idn1 45148	. . . 4 ⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   )
2		spsbc 3760	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 𝐵 = 𝐶 → [𝐴 / 𝑥]𝐵 = 𝐶))
3	1, 2	e1a 45201	. . 3 ⊢ (   𝐴 ∈ 𝑉   ▶   (∀𝑥 𝐵 = 𝐶 → [𝐴 / 𝑥]𝐵 = 𝐶)   )
4		sbceqg 4369	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶))
5	1, 4	e1a 45201	. . 3 ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)   )
6		imbi2 351	. . . 4 ⊢ (([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) → ((∀𝑥 𝐵 = 𝐶 → [𝐴 / 𝑥]𝐵 = 𝐶) ↔ (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)))
7	6	biimpcd 252	. . 3 ⊢ ((∀𝑥 𝐵 = 𝐶 → [𝐴 / 𝑥]𝐵 = 𝐶) → (([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) → (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)))
8	3, 5, 7	e11 45262	. 2 ⊢ (   𝐴 ∈ 𝑉   ▶   (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)   )
9	8	in1 45145	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1561   = wceq 1563   ∈ wcel 2145  [wsbc 3747  ⦋csb 3855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-sbc 3748  df-csb 3856  df-vd1 45144

This theorem is referenced by: (None)