Theorem bj-ceqsal

Description: Remove from ceqsal dependency on ax-ext (and on df-cleq , df-v , df-clab , df-sb ). (Contributed by BJ, 12-Oct-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	bj-ceqsal.1	⊢ Ⅎ 𝑥 𝜓
	bj-ceqsal.2	⊢ 𝐴 ∈ V
	bj-ceqsal.3	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	bj-ceqsal	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		bj-ceqsal.1	⊢ Ⅎ 𝑥 𝜓
2		bj-ceqsal.2	⊢ 𝐴 ∈ V
3		bj-ceqsal.3	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
4	1 3	bj-ceqsalgv	⊢ ( 𝐴 ∈ V → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) )
5	2 4	ax-mp	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 )