Theorem ceqsal 2750

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
ceqsal.1	⊢ Ⅎ𝑥𝜓
ceqsal.2	⊢ 𝐴 ∈ V
ceqsal.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

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

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem ceqsal

Step	Hyp	Ref	Expression
1		ceqsal.2	. 2 ⊢ 𝐴 ∈ V
2		ceqsal.1	. . 3 ⊢ Ⅎ𝑥𝜓
3		ceqsal.3	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	2, 3	ceqsalg 2749	. 2 ⊢ (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
5	1, 4	ax-mp 5	1 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1340   = wceq 1342  Ⅎwnf 1447   ∈ wcel 2135  Vcvv 2721

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 1434  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-v 2723

This theorem is referenced by:  ceqsalv  2751