Theorem vtocl 2784

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 1521	. 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 2783	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1348   ∈ wcel 2141  Vcvv 2730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by:  vtoclb  2787  zfauscl  4109  bnd2  4159  uniex  4422  ordtriexmid  4505  onsucsssucexmid  4511  regexmid  4519  ordsoexmid  4546  onintexmid  4557  reg3exmid  4564  nnregexmid  4605  acexmidlemv  5851  caovcan  6017  findcard2  6867  findcard2s  6868  inffiexmid  6884  sup3exmid  8873  bj-uniex  13952  bj-omtrans  13991