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Metamath Proof Explorer

Theorem ceqsexv2d

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016) Shorten, reduce dv conditions. (Revised by Wolf Lammen, 5-Jun-2025) (Proof shortened by SN, 5-Jun-2025)

	Ref	Expression
Hypotheses	ceqsexv2d.1	⊢ 𝐴 ∈ V
	ceqsexv2d.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	ceqsexv2d.3	⊢ 𝜓
Assertion	ceqsexv2d	⊢ ∃ 𝑥 𝜑

Proof

Step	Hyp	Ref	Expression
1		ceqsexv2d.1	⊢ 𝐴 ∈ V
2		ceqsexv2d.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3		ceqsexv2d.3	⊢ 𝜓
4	1	isseti	⊢ ∃ 𝑥 𝑥 = 𝐴
5	3 2	mpbiri	⊢ ( 𝑥 = 𝐴 → 𝜑 )
6	4 5	eximii	⊢ ∃ 𝑥 𝜑