Theorem ceqsexOLD 3541

Description: Obsolete version of ceqsex 3540 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 476	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) → 𝜓)
4	1, 3	exlimi 2218	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → 𝜓)
5	2	biimprcd 250	. . . 4 ⊢ (𝜓 → (𝑥 = 𝐴 → 𝜑))
6	1, 5	alrimi 2214	. . 3 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑))
7		ceqsex.2	. . . 4 ⊢ 𝐴 ∈ V
8	7	isseti 3506	. . 3 ⊢ ∃𝑥 𝑥 = 𝐴
9		exintr 1891	. . 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 209	1 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   = wceq 1537  ∃wex 1777  Ⅎwnf 1781   ∈ wcel 2108  Vcvv 3488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-nf 1782  df-clel 2819

This theorem is referenced by: (None)