Theorem rexlimddvcbv

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 . Usage of this theorem is discouraged because it depends on ax-13 , see rexlimddvcbvw for a weaker version that does not require it. (Contributed by Rohan Ridenour, 3-Aug-2023) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		rexlimddvcbv.1	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜃 )
2		rexlimddvcbv.2	⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐴 ∧ 𝜒 ) ) → 𝜓 )
3		rexlimddvcbv.3	⊢ ( 𝑥 = 𝑦 → ( 𝜃 ↔ 𝜒 ) )
4	3 2	rexlimdvaacbv	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜃 → 𝜓 ) )
5	1 4	mpd	⊢ ( 𝜑 → 𝜓 )