Theorem rexlimddvcbvw 40634

Description: Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 40633. The equivalent of this theorem without the bound variable change is rexlimddv 3290. Version of rexlimddvcbv 40635 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Revised by Gino Giotto, 2-Apr-2024.)

Hypotheses

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

Assertion

Ref	Expression
rexlimddvcbvw	⊢ (𝜑 → 𝜓)

Distinct variable groups:   𝜑,𝑦   𝜓,𝑦   𝜒,𝑥   𝜃,𝑦   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑥)

Proof of Theorem rexlimddvcbvw

Step	Hyp	Ref	Expression
1		rexlimddvcbvw.1	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃)
2		rexlimddvcbvw.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒))
3	2	cbvrexvw 3447	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜃 ↔ ∃𝑦 ∈ 𝐴 𝜒)
4		rexlimddvcbvw.2	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓)
5	4	rexlimdvaa 3284	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ 𝐴 𝜒 → 𝜓))
6	3, 5	syl5bi 244	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜃 → 𝜓))
7	1, 6	mpd 15	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∈ wcel 2113  ∃wrex 3138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-clel 2892  df-ral 3142  df-rex 3143

This theorem is referenced by:  mnuprdlem1  40683  mnuprdlem2  40684