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Metamath Proof Explorer

Theorem ceqsex

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995) (Revised by Mario Carneiro, 10-Oct-2016) (Proof shortened by Wolf Lammen, 22-Jan-2025)

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

Proof

Step Hyp Ref Expression
1 ceqsex.1 ⊢ Ⅎ 𝑥 𝜓
2 ceqsex.2 ⊢ 𝐴 ∈ V
3 ceqsex.3 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
4 alinexa ⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ¬ 𝜑 ) ↔ ¬ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) )
5 1 nfn ⊢ Ⅎ 𝑥 ¬ 𝜓
6 3 notbid ⊢ ( 𝑥 = 𝐴 → ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
7 5 2 6 ceqsal ⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ¬ 𝜑 ) ↔ ¬ 𝜓 )
8 4 7 bitr3i ⊢ ( ¬ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) ↔ ¬ 𝜓 )
9 8 con4bii ⊢ ( ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) ↔ 𝜓 )