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Theorem ralxp3es 8121

Description: Restricted for-all over a triple Cartesian product with explicit substitution. (Contributed by Scott Fenton, 22-Aug-2024.)

**Assertion**
Ref	Expression
ralxp3es	⊢ (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)[(1^st ‘(1^st ‘𝑥)) / 𝑦][(2^nd ‘(1^st ‘𝑥)) / 𝑧][(2^nd ‘𝑥) / 𝑤]𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜑)

Distinct variable groups: 𝑤,𝐴,𝑥,𝑦,𝑧 𝑤,𝐵,𝑥,𝑦,𝑧 𝑤,𝐶,𝑥,𝑦,𝑧 𝜑,𝑥 Allowed substitution hints: 𝜑(𝑦,𝑧,𝑤)

**Proof of Theorem ralxp3es**
Step	Hyp	Ref	Expression
1		nfsbc1v 3766	. 2 ⊢ Ⅎ𝑦[(1^st ‘(1^st ‘𝑥)) / 𝑦][(2^nd ‘(1^st ‘𝑥)) / 𝑧][(2^nd ‘𝑥) / 𝑤]𝜑
2		nfcv 2926	. . 3 ⊢ Ⅎ𝑧(1^st ‘(1^st ‘𝑥))
3		nfsbc1v 3766	. . 3 ⊢ Ⅎ𝑧[(2^nd ‘(1^st ‘𝑥)) / 𝑧][(2^nd ‘𝑥) / 𝑤]𝜑
4	2, 3	nfsbcw 3768	. 2 ⊢ Ⅎ𝑧[(1^st ‘(1^st ‘𝑥)) / 𝑦][(2^nd ‘(1^st ‘𝑥)) / 𝑧][(2^nd ‘𝑥) / 𝑤]𝜑
5		nfcv 2926	. . 3 ⊢ Ⅎ𝑤(1^st ‘(1^st ‘𝑥))
6		nfcv 2926	. . . 4 ⊢ Ⅎ𝑤(2^nd ‘(1^st ‘𝑥))
7		nfsbc1v 3766	. . . 4 ⊢ Ⅎ𝑤[(2^nd ‘𝑥) / 𝑤]𝜑
8	6, 7	nfsbcw 3768	. . 3 ⊢ Ⅎ𝑤[(2^nd ‘(1^st ‘𝑥)) / 𝑧][(2^nd ‘𝑥) / 𝑤]𝜑
9	5, 8	nfsbcw 3768	. 2 ⊢ Ⅎ𝑤[(1^st ‘(1^st ‘𝑥)) / 𝑦][(2^nd ‘(1^st ‘𝑥)) / 𝑧][(2^nd ‘𝑥) / 𝑤]𝜑
10		nfv 1936	. 2 ⊢ Ⅎ𝑥𝜑
11		sbcoteq1a 8034	. 2 ⊢ (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → ([(1^st ‘(1^st ‘𝑥)) / 𝑦][(2^nd ‘(1^st ‘𝑥)) / 𝑧][(2^nd ‘𝑥) / 𝑤]𝜑 ↔ 𝜑))
12	1, 4, 9, 10, 11	ralxp3f 8119	1 ⊢ (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)[(1^st ‘(1^st ‘𝑥)) / 𝑦][(2^nd ‘(1^st ‘𝑥)) / 𝑧][(2^nd ‘𝑥) / 𝑤]𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜑)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 208 ∀wral 3078 [wsbc 3746 × cxp 5647 ‘cfv 6523 1^st c1st 7970 2^nd c2nd 7971

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-sep 5248 ax-nul 5258 ax-pr 5392 ax-un 7720

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3079 df-rex 3089 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-ot 4593 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5544 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-iota 6479 df-fun 6525 df-fv 6531 df-1st 7972 df-2nd 7973

This theorem is referenced by: frpoins3xp3g 8123