Theorem rexlimddvcbvw 44202

Description: Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 44201. The equivalent of this theorem without the bound variable change is rexlimddv 3141. Version of rexlimddvcbv 44203 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Revised by GG, 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 3217	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜃 ↔ ∃𝑦 ∈ 𝐴 𝜒)
4		rexlimddvcbvw.2	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓)
5	4	rexlimdvaa 3136	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ 𝐴 𝜒 → 𝜓))
6	3, 5	biimtrid 242	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜃 → 𝜓))
7	1, 6	mpd 15	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  ∃wrex 3054

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 1967  ax-7 2008  ax-8 2111

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-clel 2804  df-rex 3055

This theorem is referenced by:  mnuprdlem1  44268  mnuprdlem2  44269