Theorem vtocl 2674

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 1467	. 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 2673	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1290   ∈ wcel 1439  Vcvv 2620

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-v 2622

This theorem is referenced by:  vtoclb  2677  zfauscl  3965  bnd2  4014  uniex  4273  ordtriexmid  4351  onsucsssucexmid  4356  regexmid  4364  ordsoexmid  4391  onintexmid  4401  reg3exmid  4408  nnregexmid  4447  acexmidlemv  5664  caovcan  5823  findcard2  6659  findcard2s  6660  inffiexmid  6676  bj-uniex  12081  bj-omtrans  12124