Theorem bj-ceqsalv 37557

Description: Remove from ceqsalv 3494 dependency on ax-ext 2735 (and on df-cleq 2755, df-v 3457, df-clab 2742, df-sb 2097). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)

Hypotheses

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

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-ceqsalv

Step	Hyp	Ref	Expression
1		nfv 1944	. 2 ⊢ Ⅎ𝑥𝜓
2		bj-ceqsalv.1	. 2 ⊢ 𝐴 ∈ V
3		bj-ceqsalv.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	bj-ceqsal 37556	1 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570   ∈ wcel 2143  Vcvv 3455

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-12 2213

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1105  df-ex 1810  df-nf 1814  df-clel 2838

This theorem is used by: (None)