Theorem rexlimddvcbvw

Description: Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv . The equivalent of this theorem without the bound variable change is rexlimddv . Version of rexlimddvcbv with a disjoint variable condition, which does not require ax-13 . (Contributed by Rohan Ridenour, 3-Aug-2023) (Revised by Gino Giotto, 2-Apr-2024)

	Ref	Expression
Hypotheses	rexlimddvcbvw.1	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜃 )
	rexlimddvcbvw.2	⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐴 ∧ 𝜒 ) ) → 𝜓 )
	rexlimddvcbvw.3	⊢ ( 𝑥 = 𝑦 → ( 𝜃 ↔ 𝜒 ) )
Assertion	rexlimddvcbvw	⊢ ( 𝜑 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		rexlimddvcbvw.1	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜃 )
2		rexlimddvcbvw.2	⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐴 ∧ 𝜒 ) ) → 𝜓 )
3		rexlimddvcbvw.3	⊢ ( 𝑥 = 𝑦 → ( 𝜃 ↔ 𝜒 ) )
4	3	cbvrexvw	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜃 ↔ ∃ 𝑦 ∈ 𝐴 𝜒 )
5	2	rexlimdvaa	⊢ ( 𝜑 → ( ∃ 𝑦 ∈ 𝐴 𝜒 → 𝜓 ) )
6	4 5	syl5bi	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜃 → 𝜓 ) )
7	1 6	mpd	⊢ ( 𝜑 → 𝜓 )