Theorem ceqsexOLD 3524

Description: Obsolete version of ceqsex 3523 as of 22-Jan-2025. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
ceqsex.1	⊢ Ⅎ𝑥𝜓
ceqsex.2	⊢ 𝐴 ∈ V
ceqsex.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ceqsexOLD	⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem ceqsexOLD

Step	Hyp	Ref	Expression
1		ceqsex.1	. . 3 ⊢ Ⅎ𝑥𝜓
2		ceqsex.3	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	2	biimpa 478	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) → 𝜓)
4	1, 3	exlimi 2211	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → 𝜓)
5	2	biimprcd 249	. . . 4 ⊢ (𝜓 → (𝑥 = 𝐴 → 𝜑))
6	1, 5	alrimi 2207	. . 3 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑))
7		ceqsex.2	. . . 4 ⊢ 𝐴 ∈ V
8	7	isseti 3490	. . 3 ⊢ ∃𝑥 𝑥 = 𝐴
9		exintr 1896	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
10	6, 8, 9	mpisyl 21	. 2 ⊢ (𝜓 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
11	4, 10	impbii 208	1 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397  ∀wal 1540   = wceq 1542  ∃wex 1782  Ⅎwnf 1786   ∈ wcel 2107  Vcvv 3475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-12 2172

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-nf 1787  df-clel 2811

This theorem is referenced by: (None)