Metamath Proof Explorer

Theorem cgsex4gOLD

Description: Obsolete version of cgsex4g as of 28-Jun-2024. (Contributed by NM, 5-Aug-1995) Avoid ax-10 . (Revised by Gino Giotto, 20-Aug-2023) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses cgsex4gOLD.1 ⊢ ( ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) → 𝜒 )
cgsex4gOLD.2 ⊢ ( 𝜒 → ( 𝜑 ↔ 𝜓 ) )
Assertion cgsex4gOLD ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) ↔ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 cgsex4gOLD.1 ⊢ ( ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) → 𝜒 )
2 cgsex4gOLD.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 exdistrv ⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) )
10 8 9 sylibr ⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) )
11 elisset ⊢ ( 𝐶 ∈ 𝑅 → ∃ 𝑧 𝑧 = 𝐶 )
12 elisset ⊢ ( 𝐷 ∈ 𝑆 → ∃ 𝑤 𝑤 = 𝐷 )
13 11 12 anim12i ⊢ ( ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) → ( ∃ 𝑧 𝑧 = 𝐶 ∧ ∃ 𝑤 𝑤 = 𝐷 ) )
14 exdistrv ⊢ ( ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ↔ ( ∃ 𝑧 𝑧 = 𝐶 ∧ ∃ 𝑤 𝑤 = 𝐷 ) )
15 13 14 sylibr ⊢ ( ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) → ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) )
16 10 15 anim12i ⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
17 excom ⊢ ( ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ∃ 𝑧 ∃ 𝑦 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
18 17 exbii ⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ∃ 𝑥 ∃ 𝑧 ∃ 𝑦 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
19 4exdistrv ⊢ ( ∃ 𝑥 ∃ 𝑧 ∃ 𝑦 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
20 18 19 bitri ⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
21 16 20 sylibr ⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
22 1 2eximi ⊢ ( ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) → ∃ 𝑧 ∃ 𝑤 𝜒 )
23 22 2eximi ⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) → ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 𝜒 )
24 21 23 syl ⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 𝜒 )
25 2 biimprcd ⊢ ( 𝜓 → ( 𝜒 → 𝜑 ) )
26 25 ancld ⊢ ( 𝜓 → ( 𝜒 → ( 𝜒 ∧ 𝜑 ) ) )
27 26 2eximdv ⊢ ( 𝜓 → ( ∃ 𝑧 ∃ 𝑤 𝜒 → ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) ) )
28 27 2eximdv ⊢ ( 𝜓 → ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 𝜒 → ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) ) )
29 24 28 syl5com ⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ( 𝜓 → ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) ) )
30 5 29 impbid2 ⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) ↔ 𝜓 ) )