Theorem ceqsalgALT 3473

Description: Alternate proof of ceqsalg 3472, not using ceqsalt 3470. (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.)

Hypotheses

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

Assertion

Ref	Expression
ceqsalgALT	⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem ceqsalgALT

Step	Hyp	Ref	Expression
1		elisset 2813	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		nfa1 2154	. . . 4 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝐴 → 𝜑)
3		ceqsalg.1	. . . 4 ⊢ Ⅎ𝑥𝜓
4		ceqsalg.2	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	4	biimpd 229	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
6	5	a2i 14	. . . . 5 ⊢ ((𝑥 = 𝐴 → 𝜑) → (𝑥 = 𝐴 → 𝜓))
7	6	sps 2188	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (𝑥 = 𝐴 → 𝜓))
8	2, 3, 7	exlimd 2221	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (∃𝑥 𝑥 = 𝐴 → 𝜓))
9	1, 8	syl5com 31	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) → 𝜓))
10	4	biimprcd 250	. . 3 ⊢ (𝜓 → (𝑥 = 𝐴 → 𝜑))
11	3, 10	alrimi 2216	. 2 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑))
12	9, 11	impbid1 225	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1539   = wceq 1541  ∃wex 1780  Ⅎwnf 1784   ∈ wcel 2111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-10 2144  ax-12 2180

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-clel 2806

This theorem is referenced by: (None)