Theorem ceqsexvOLD 3456

Description: Obsolete version of ceqsexv 3455 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 1922	. 2 ⊢ Ⅎ𝑥𝜓
2		ceqsexvOLD.1	. 2 ⊢ 𝐴 ∈ V
3		ceqsexvOLD.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	ceqsex 3454	1 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543  ∃wex 1787   ∈ wcel 2110  Vcvv 3408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-nf 1792  df-clel 2816

This theorem is referenced by: (None)