Theorem ceqsexv

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995) Avoid ax-12 . (Revised by Gino Giotto, 12-Oct-2024) (Proof shortened by Wolf Lammen, 22-Jan-2025)

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

Proof

Step	Hyp	Ref	Expression
1		ceqsexv.1	⊢ 𝐴 ∈ V
2		ceqsexv.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3		alinexa	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ¬ 𝜑 ) ↔ ¬ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) )
4	2	notbid	⊢ ( 𝑥 = 𝐴 → ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
5	1 4	ceqsalv	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ¬ 𝜑 ) ↔ ¬ 𝜓 )
6	3 5	bitr3i	⊢ ( ¬ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) ↔ ¬ 𝜓 )
7	6	con4bii	⊢ ( ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) ↔ 𝜓 )