Theorem ceqsexv2dOLD 3546

Description: Obsolete version of ceqsexv2d 3545 as of 5-Jun-2025. (Contributed by Thierry Arnoux, 10-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ceqsexv2dOLD.1	⊢ 𝐴 ∈ V
ceqsexv2dOLD.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
ceqsexv2dOLD.3	⊢ 𝜓

Assertion

Ref	Expression
ceqsexv2dOLD	⊢ ∃𝑥𝜑

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqsexv2dOLD

Step	Hyp	Ref	Expression
1		ceqsexv2dOLD.3	. 2 ⊢ 𝜓
2		ceqsexv2dOLD.1	. . . 4 ⊢ 𝐴 ∈ V
3		ceqsexv2dOLD.2	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	2, 3	ceqsexv 3542	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)
5	4	biimpri 228	. 2 ⊢ (𝜓 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
6		exsimpr 1868	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → ∃𝑥𝜑)
7	1, 5, 6	mp2b 10	1 ⊢ ∃𝑥𝜑

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ 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

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

This theorem is referenced by: (None)