Theorem ceqsalgALT

Description: Alternate proof of ceqsalg , not using ceqsalt . (Contributed by NM, 29-Oct-2003) (Proof shortened by Andrew Salmon, 8-Jun-2011) (Revised by BJ, 29-Sep-2019) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ceqsalg.1	⊢ Ⅎ 𝑥 𝜓
	ceqsalg.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	ceqsalgALT	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		ceqsalg.1	⊢ Ⅎ 𝑥 𝜓
2		ceqsalg.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3		elisset	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 )
4		nfa1	⊢ Ⅎ 𝑥 ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 )
5	2	biimpd	⊢ ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) )
6	5	a2i	⊢ ( ( 𝑥 = 𝐴 → 𝜑 ) → ( 𝑥 = 𝐴 → 𝜓 ) )
7	6	sps	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) → ( 𝑥 = 𝐴 → 𝜓 ) )
8	4 1 7	exlimd	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) → ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) )
9	3 8	syl5com	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) → 𝜓 ) )
10	2	biimprcd	⊢ ( 𝜓 → ( 𝑥 = 𝐴 → 𝜑 ) )
11	1 10	alrimi	⊢ ( 𝜓 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) )
12	9 11	impbid1	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) )