Theorem vtocl 1378

Description: Implicit substitution of a class for a set variable.

Hypotheses

Ref	Expression
vtocl.1	⊢ A ∈ V
vtocl.2	⊢ (x = A → (φ ↔ ψ))
vtocl.3	⊢ φ

Assertion

Ref	Expression
vtocl	⊢ ψ

Distinct variable group(s):   x,A   ψ,x

Proof of Theorem vtocl

Step	Hyp	Ref	Expression
1		ax-17 925	. 2 ⊢ (ψ → ∀xψ)
2		vtocl.1	. 2 ⊢ A ∈ V
3		vtocl.2	. 2 ⊢ (x = A → (φ ↔ ψ))
4		vtocl.3	. 2 ⊢ φ
5	1, 2, 3, 4	vtoclf 1377	1 ⊢ ψ

Syntax hints:   → wi 2   ↔ wb 127   = wceq 1091   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  vtoclb 1381  axrep 1473  pwex 1806  uniex 1947  fnfvbr 2855  caoprcan 3069  zfregcl 3446  bnd2 3549  ac4c 3572  ac5 3573  kmlem2 3581

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

metamath.org