Theorem vtoclga 2921

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 20-Aug-1995.)

Hypotheses

Ref	Expression
vtoclga.1	⊢ (x = A → (φ ↔ ψ))
vtoclga.2	⊢ (x ∈ B → φ)

Assertion

Ref	Expression
vtoclga	⊢ (A ∈ B → ψ)

Distinct variable groups:   x,A   x,B   ψ,x

Allowed substitution hint:   φ(x)

Proof of Theorem vtoclga

Step	Hyp	Ref	Expression
1		nfcv 2490	. 2 ⊢ ℲxA
2		nfv 1619	. 2 ⊢ Ⅎxψ
3		vtoclga.1	. 2 ⊢ (x = A → (φ ↔ ψ))
4		vtoclga.2	. 2 ⊢ (x ∈ B → φ)
5	1, 2, 3, 4	vtoclgaf 2920	1 ⊢ (A ∈ B → 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:  vtoclri  2930  ssuni  3914  peano2  4404  nncaddccl  4420  ltfintri  4467  ssfin  4471  nnadjoin  4521  fneu  5188  fnressn  5439  fressnfv  5440  ndmovcl  5615  caovord  5630  caovmo  5646  fvmptss  5706  leconnnc  6219