Theorem cgsex2gd 37386

Description: Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.) Adapt cgsex2g 3488 $p to deduction form. (Revised by BJ, 28-Mar-2026.) Do not use cgsex2g 3488. (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 1472	. . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
3	2	3expib 1123	. . . 4 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))
4	3	exlimdvv 1936	. . 3 ⊢ (𝜑 → (∃𝑥∃𝑦(𝜓 ∧ 𝜒) → 𝜃))
5	4	adantr 480	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (∃𝑥∃𝑦(𝜓 ∧ 𝜒) → 𝜃))
6		cgsex2gd.is	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝜓)
7	6	ex 412	. . . . 5 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝜓))
8	7	2eximdv 1921	. . . 4 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑥∃𝑦𝜓))
9		elisset 2819	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
10		elisset 2819	. . . . . 6 ⊢ (𝐵 ∈ 𝑊 → ∃𝑦 𝑦 = 𝐵)
11	9, 10	anim12i 614	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
12		exdistrv 1957	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
13	11, 12	sylibr 234	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
14	8, 13	impel 505	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ∃𝑥∃𝑦𝜓)
15	1	biimprd 248	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝜃 → 𝜒))
16	15	impancom 451	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → 𝜒))
17	16	ancld 550	. . . . . 6 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → (𝜓 ∧ 𝜒)))
18	17	2eximdv 1921	. . . . 5 ⊢ ((𝜑 ∧ 𝜃) → (∃𝑥∃𝑦𝜓 → ∃𝑥∃𝑦(𝜓 ∧ 𝜒)))
19	18	expimpd 453	. . . 4 ⊢ (𝜑 → ((𝜃 ∧ ∃𝑥∃𝑦𝜓) → ∃𝑥∃𝑦(𝜓 ∧ 𝜒)))
20	19	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((𝜃 ∧ ∃𝑥∃𝑦𝜓) → ∃𝑥∃𝑦(𝜓 ∧ 𝜒)))
21	14, 20	mpan2d 695	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (𝜃 → ∃𝑥∃𝑦(𝜓 ∧ 𝜒)))
22	5, 21	impbid 212	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (∃𝑥∃𝑦(𝜓 ∧ 𝜒) ↔ 𝜃))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-clel 2812

This theorem is referenced by:  copsex2gd  37387