Metamath Proof Explorer

Theorem cgsex4g

Description: An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995) Avoid ax-10 , ax-11 . (Revised by Gino Giotto, 28-Jun-2024) Avoid ax-9 , ax-ext . (Revised by Wolf Lammen, 21-Mar-2025)

Ref Expression
Hypotheses cgsex4g.1 ⊢ ( ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) → 𝜒 )
cgsex4g.2 ⊢ ( 𝜒 → ( 𝜑 ↔ 𝜓 ) )
Assertion cgsex4g ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) ↔ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 cgsex4g.1 ⊢ ( ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) → 𝜒 )
2 cgsex4g.2 ⊢ ( 𝜒 → ( 𝜑 ↔ 𝜓 ) )
3 2 biimpa ⊢ ( ( 𝜒 ∧ 𝜑 ) → 𝜓 )
4 3 exlimivv ⊢ ( ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) → 𝜓 )
5 4 exlimivv ⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) → 𝜓 )
6 elisset ⊢ ( 𝐴 ∈ 𝑅 → ∃ 𝑥 𝑥 = 𝐴 )
7 elisset ⊢ ( 𝐵 ∈ 𝑆 → ∃ 𝑦 𝑦 = 𝐵 )
8 6 7 anim12i ⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) )
9 elisset ⊢ ( 𝐶 ∈ 𝑅 → ∃ 𝑧 𝑧 = 𝐶 )
10 elisset ⊢ ( 𝐷 ∈ 𝑆 → ∃ 𝑤 𝑤 = 𝐷 )
11 9 10 anim12i ⊢ ( ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) → ( ∃ 𝑧 𝑧 = 𝐶 ∧ ∃ 𝑤 𝑤 = 𝐷 ) )
12 8 11 anim12i ⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ( ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) ∧ ( ∃ 𝑧 𝑧 = 𝐶 ∧ ∃ 𝑤 𝑤 = 𝐷 ) ) )
13 19.42vv ⊢ ( ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
14 13 2exbii ⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ∃ 𝑥 ∃ 𝑦 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
15 19.41vv ⊢ ( ∃ 𝑥 ∃ 𝑦 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
16 exdistrv ⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) )
17 exdistrv ⊢ ( ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ↔ ( ∃ 𝑧 𝑧 = 𝐶 ∧ ∃ 𝑤 𝑤 = 𝐷 ) )
18 16 17 anbi12i ⊢ ( ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ( ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) ∧ ( ∃ 𝑧 𝑧 = 𝐶 ∧ ∃ 𝑤 𝑤 = 𝐷 ) ) )
19 14 15 18 3bitri ⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ( ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) ∧ ( ∃ 𝑧 𝑧 = 𝐶 ∧ ∃ 𝑤 𝑤 = 𝐷 ) ) )
20 12 19 sylibr ⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
21 1 2eximi ⊢ ( ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) → ∃ 𝑧 ∃ 𝑤 𝜒 )
22 21 2eximi ⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) → ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 𝜒 )
23 20 22 syl ⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 𝜒 )
24 2 biimprcd ⊢ ( 𝜓 → ( 𝜒 → 𝜑 ) )
25 24 ancld ⊢ ( 𝜓 → ( 𝜒 → ( 𝜒 ∧ 𝜑 ) ) )
26 25 2eximdv ⊢ ( 𝜓 → ( ∃ 𝑧 ∃ 𝑤 𝜒 → ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) ) )
27 26 2eximdv ⊢ ( 𝜓 → ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 𝜒 → ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) ) )
28 23 27 syl5com ⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ( 𝜓 → ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) ) )
29 5 28 impbid2 ⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) ↔ 𝜓 ) )