Theorem vtocl 2793

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 30-Aug-1993.)

Hypotheses

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

Assertion

Ref	Expression
vtocl	⊢ 𝜓

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem vtocl

Step	Hyp	Ref	Expression
1		nfv 1528	. 2 ⊢ Ⅎ𝑥𝜓
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 2792	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1353   ∈ wcel 2148  Vcvv 2739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2741

This theorem is referenced by:  vtoclb  2796  zfauscl  4125  bnd2  4175  uniex  4439  ordtriexmid  4522  onsucsssucexmid  4528  regexmid  4536  ordsoexmid  4563  onintexmid  4574  reg3exmid  4581  nnregexmid  4622  acexmidlemv  5875  caovcan  6041  findcard2  6891  findcard2s  6892  inffiexmid  6908  sup3exmid  8916  bj-uniex  14754  bj-omtrans  14793