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Metamath Proof Explorer

Theorem rexbidva

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 9-Mar-1997) Reduce dependencies on axioms. (Revised by Wolf Lammen, 6-Dec-2019) (Proof shortened by Wolf Lammen, 10-Dec-2019)

		Ref	Expression
	Hypothesis	ralbidva.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
	Assertion	rexbidva	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ∈ 𝐴 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		ralbidva.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
2	1	pm5.32da	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜒 ) ) )
3	2	rexbidv2	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ∈ 𝐴 𝜒 ) )