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Metamath Proof Explorer

Theorem rexeqbii

Description: Equality deduction for restricted existential quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017)

	Ref	Expression
Hypotheses	rexeqbii.1	⊢ 𝐴 = 𝐵
	rexeqbii.2	⊢ ( 𝜓 ↔ 𝜒 )
Assertion	rexeqbii	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ∈ 𝐵 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		rexeqbii.1	⊢ 𝐴 = 𝐵
2		rexeqbii.2	⊢ ( 𝜓 ↔ 𝜒 )
3	1	eleq2i	⊢ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 )
4	3 2	anbi12i	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜒 ) )
5	4	rexbii2	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ∈ 𝐵 𝜒 )