Theorem cgsex2gd 37510

Description: Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.) Adapt cgsex2g 3478 to deduction form. (Revised by BJ, 28-Mar-2026.) Do not use cgsex2g 3478. (Proof modification is discouraged.)

Hypotheses

Ref	Expression
cgsex2gd.is	⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝜓)
cgsex2gd.maj	⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))

Assertion

Ref	Expression
cgsex2gd	⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (∃𝑥∃𝑦(𝜓 ∧ 𝜒) ↔ 𝜃))

Distinct variable groups:   𝜑,𝑥,𝑦   𝜃,𝑥,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem cgsex2gd

Step	Hyp	Ref	Expression
1		cgsex2gd.maj	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
2	1	biimp3a 1478	. . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
3	2	3expib 1129	. . . 4 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))
4	3	exlimdvv 1942	. . 3 ⊢ (𝜑 → (∃𝑥∃𝑦(𝜓 ∧ 𝜒) → 𝜃))
5	4	adantr 482	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (∃𝑥∃𝑦(𝜓 ∧ 𝜒) → 𝜃))
6		cgsex2gd.is	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝜓)
7	6	ex 414	. . . . 5 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝜓))
8	7	2eximdv 1927	. . . 4 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑥∃𝑦𝜓))
9		elisset 2823	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
10		elisset 2823	. . . . . 6 ⊢ (𝐵 ∈ 𝑊 → ∃𝑦 𝑦 = 𝐵)
11	9, 10	anim12i 620	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
12		exdistrv 1963	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
13	11, 12	sylibr 236	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
14	8, 13	impel 511	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ∃𝑥∃𝑦𝜓)
15	1	biimprd 250	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝜃 → 𝜒))
16	15	impancom 453	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → 𝜒))
17	16	ancld 556	. . . . . 6 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → (𝜓 ∧ 𝜒)))
18	17	2eximdv 1927	. . . . 5 ⊢ ((𝜑 ∧ 𝜃) → (∃𝑥∃𝑦𝜓 → ∃𝑥∃𝑦(𝜓 ∧ 𝜒)))
19	18	expimpd 455	. . . 4 ⊢ (𝜑 → ((𝜃 ∧ ∃𝑥∃𝑦𝜓) → ∃𝑥∃𝑦(𝜓 ∧ 𝜒)))
20	19	adantr 482	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((𝜃 ∧ ∃𝑥∃𝑦𝜓) → ∃𝑥∃𝑦(𝜓 ∧ 𝜒)))
21	14, 20	mpan2d 701	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (𝜃 → ∃𝑥∃𝑦(𝜓 ∧ 𝜒)))
22	5, 21	impbid 214	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (∃𝑥∃𝑦(𝜓 ∧ 𝜒) ↔ 𝜃))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548  ∃wex 1787   ∈ wcel 2121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123

This theorem depends on definitions:  df-bi 209  df-an 398  df-3an 1095  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-clel 2816

This theorem is referenced by:  copsex2gd  37511