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Metamath Proof Explorer

Theorem rexeqf

Description: Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. See rexeq for a version based on fewer axioms. (Contributed by NM, 9-Oct-2003) (Revised by Andrew Salmon, 11-Jul-2011) (Proof shortened by Wolf Lammen, 9-Mar-2025)

Ref Expression
Hypotheses raleqf.1 ⊢ Ⅎ 𝑥 𝐴
raleqf.2 ⊢ Ⅎ 𝑥 𝐵
Assertion rexeqf ( 𝐴 = 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ∈ 𝐵 𝜑 ) )

Proof

Step Hyp Ref Expression
1 raleqf.1 ⊢ Ⅎ 𝑥 𝐴
2 raleqf.2 ⊢ Ⅎ 𝑥 𝐵
3 1 2 raleqf ⊢ ( 𝐴 = 𝐵 → ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀ 𝑥 ∈ 𝐵 ¬ 𝜑 ) )
4 ralnex ⊢ ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃ 𝑥 ∈ 𝐴 𝜑 )
5 ralnex ⊢ ( ∀ 𝑥 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃ 𝑥 ∈ 𝐵 𝜑 )
6 3 4 5 3bitr3g ⊢ ( 𝐴 = 𝐵 → ( ¬ ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃ 𝑥 ∈ 𝐵 𝜑 ) )
7 6 con4bid ⊢ ( 𝐴 = 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ∈ 𝐵 𝜑 ) )