Theorem vtoclg 2915

Description: Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995.)

Hypotheses

Ref	Expression
vtoclg.1	⊢ (x = A → (φ ↔ ψ))
vtoclg.2	⊢ φ

Assertion

Ref	Expression
vtoclg	⊢ (A ∈ V → ψ)

Distinct variable groups:   x,A   ψ,x

Allowed substitution hints:   φ(x)   V(x)

Proof of Theorem vtoclg

Step	Hyp	Ref	Expression
1		nfcv 2490	. 2 ⊢ ℲxA
2		nfv 1619	. 2 ⊢ Ⅎxψ
3		vtoclg.1	. 2 ⊢ (x = A → (φ ↔ ψ))
4		vtoclg.2	. 2 ⊢ φ
5	1, 2, 3, 4	vtoclgf 2914	1 ⊢ (A ∈ V → ψ)

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is referenced by:  vtoclbg  2916  ceqex  2970  moeq3  3014  mo2icl  3016  sbctt  3109  sbcnestgf  3184  csbing  3463  csbifg  3691  prnzg  3837  sneqrg  3875  unisng  3909  snex  4112  snel1cg  4142  xpkvexg  4286  cnvkexg  4287  p6exg  4291  sikexg  4297  ins2kexg  4306  ins3kexg  4307  iota5  4360  csbiotag  4372  ssfin  4471  csbopabg  4638  vtoclr  4817  csbima12g  4956  dmsnopg  5067  fconstg  5252  fvelimab  5371  fvi  5443  csbovg  5553  trtxp  5782  oqelins4  5795  fnfullfunlem1  5857  fvfullfun  5865  fundmeng  6045  df1c3g  6142  sbthlem2  6205  frecxp  6315  frecxpg  6316