Theorem bj-ceqsalgvALT 36915

Description: Alternate proof of bj-ceqsalgv 36914. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bj-ceqsalgv.1	⊢ Ⅎ𝑥𝜓
bj-ceqsalgv.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
bj-ceqsalgvALT	⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

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

Proof of Theorem bj-ceqsalgvALT

Step	Hyp	Ref	Expression
1		bj-ceqsalgv.1	. 2 ⊢ Ⅎ𝑥𝜓
2		bj-ceqsalgv.2	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	2	ax-gen 1795	. 2 ⊢ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4		bj-ceqsaltv 36910	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
5	1, 3, 4	mp3an12 1453	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-ex 1780  df-nf 1784  df-clel 2810

This theorem is referenced by: (None)