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Metamath Proof Explorer

Theorem rexbid

Description: Formula-building rule for restricted existential quantifier (deduction form). For a version based on fewer axioms see rexbidv . (Contributed by NM, 27-Jun-1998)

	Ref	Expression
Hypotheses	rexbid.1	⊢ Ⅎ 𝑥 𝜑
	rexbid.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	rexbid	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ∈ 𝐴 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		rexbid.1	⊢ Ⅎ 𝑥 𝜑
2		rexbid.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
3	2	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
4	1 3	rexbida	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ∈ 𝐴 𝜒 ) )