Theorem rexxfr3d

Description: Transfer existential quantification from a variable x to another variable y contained in expression A . (Contributed by SN, 20-Jun-2025)

	Ref	Expression
Hypotheses	rexxfr3d.s	⊢ ( 𝑥 = 𝑋 → ( 𝜓 ↔ 𝜒 ) )
	rexxfr3d.x	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↔ ∃ 𝑦 ∈ 𝐵 𝑥 = 𝑋 ) )
	rexxfr3d.a	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
Assertion	rexxfr3d	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑦 ∈ 𝐵 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		rexxfr3d.s	⊢ ( 𝑥 = 𝑋 → ( 𝜓 ↔ 𝜒 ) )
2		rexxfr3d.x	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↔ ∃ 𝑦 ∈ 𝐵 𝑥 = 𝑋 ) )
3		rexxfr3d.a	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
4	3	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 𝑋 ∈ 𝑉 )
5	1	adantl	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ( 𝜓 ↔ 𝜒 ) )
6	4 2 5	rexxfr2d	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑦 ∈ 𝐵 𝜒 ) )