Theorem ceqsexvOLD 3480

Description: Obsolete version of ceqsexv 3479 as of 12-Oct-2024. (Contributed by NM, 2-Mar-1995.) (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		nfv 1917	. 2 ⊢ Ⅎ𝑥𝜓
2		ceqsexvOLD.1	. 2 ⊢ 𝐴 ∈ V
3		ceqsexvOLD.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	ceqsex 3478	1 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  Vcvv 3432

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-nf 1787  df-clel 2816

This theorem is referenced by: (None)