Theorem ceqsalv 2843

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)

Hypotheses

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

Assertion

Ref	Expression
ceqsalv	⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqsalv

Step	Hyp	Ref	Expression
1		nfv 1577	. 2 ⊢ Ⅎ𝑥𝜓
2		ceqsalv.1	. 2 ⊢ 𝐴 ∈ V
3		ceqsalv.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	ceqsal 2842	1 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1396   = wceq 1398   ∈ wcel 2203  Vcvv 2812

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-v 2814

This theorem is referenced by:  gencbval  2862  clel2  2949  clel4  2952  reu8  3012  raliunxp  4895  fv3  5692