Theorem rexlimddvcbvw 44168

Description: Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 44167. The equivalent of this theorem without the bound variable change is rexlimddv 3167. Version of rexlimddvcbv 44169 with a disjoint variable condition, which does not require ax-13 2380. (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 3244	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜃 ↔ ∃𝑦 ∈ 𝐴 𝜒)
4		rexlimddvcbvw.2	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓)
5	4	rexlimdvaa 3162	. . 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 2108  ∃wrex 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-clel 2819  df-rex 3077

This theorem is referenced by:  mnuprdlem1  44241  mnuprdlem2  44242