Theorem ceqsexvOLD 3522

Description: Obsolete version of ceqsexv 3521 as of 12-Oct-2024. (Contributed by NM, 2-Mar-1995.) Avoid ax-12 2164. (Revised by Gino Giotto, 12-Oct-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqsexvOLD

Step	Hyp	Ref	Expression
1		ceqsexvOLD.2	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2	1	biimpa 476	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) → 𝜓)
3	2	exlimiv 1926	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → 𝜓)
4	1	biimprcd 249	. . . 4 ⊢ (𝜓 → (𝑥 = 𝐴 → 𝜑))
5	4	alrimiv 1923	. . 3 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑))
6		ceqsexvOLD.1	. . . 4 ⊢ 𝐴 ∈ V
7	6	isseti 3485	. . 3 ⊢ ∃𝑥 𝑥 = 𝐴
8		exintr 1888	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
9	5, 7, 8	mpisyl 21	. 2 ⊢ (𝜓 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
10	3, 9	impbii 208	1 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1532   = wceq 1534  ∃wex 1774   ∈ wcel 2099  Vcvv 3469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-clel 2805

This theorem is referenced by: (None)