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Theorem ralxp3es 8180

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 3824	. 2 ⊢ Ⅎ𝑦[(1^st ‘(1^st ‘𝑥)) / 𝑦][(2^nd ‘(1^st ‘𝑥)) / 𝑧][(2^nd ‘𝑥) / 𝑤]𝜑
2		nfcv 2908	. . 3 ⊢ Ⅎ𝑧(1^st ‘(1^st ‘𝑥))
3		nfsbc1v 3824	. . 3 ⊢ Ⅎ𝑧[(2^nd ‘(1^st ‘𝑥)) / 𝑧][(2^nd ‘𝑥) / 𝑤]𝜑
4	2, 3	nfsbcw 3826	. 2 ⊢ Ⅎ𝑧[(1^st ‘(1^st ‘𝑥)) / 𝑦][(2^nd ‘(1^st ‘𝑥)) / 𝑧][(2^nd ‘𝑥) / 𝑤]𝜑
5		nfcv 2908	. . 3 ⊢ Ⅎ𝑤(1^st ‘(1^st ‘𝑥))
6		nfcv 2908	. . . 4 ⊢ Ⅎ𝑤(2^nd ‘(1^st ‘𝑥))
7		nfsbc1v 3824	. . . 4 ⊢ Ⅎ𝑤[(2^nd ‘𝑥) / 𝑤]𝜑
8	6, 7	nfsbcw 3826	. . 3 ⊢ Ⅎ𝑤[(2^nd ‘(1^st ‘𝑥)) / 𝑧][(2^nd ‘𝑥) / 𝑤]𝜑
9	5, 8	nfsbcw 3826	. 2 ⊢ Ⅎ𝑤[(1^st ‘(1^st ‘𝑥)) / 𝑦][(2^nd ‘(1^st ‘𝑥)) / 𝑧][(2^nd ‘𝑥) / 𝑤]𝜑
10		nfv 1913	. 2 ⊢ Ⅎ𝑥𝜑
11		sbcoteq1a 8092	. 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 8178	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 206 ∀wral 3067 [wsbc 3804 × cxp 5698 ‘cfv 6573 1^st c1st 8028 2^nd c2nd 8029

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-ot 4657 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-iota 6525 df-fun 6575 df-fv 6581 df-1st 8030 df-2nd 8031

This theorem is referenced by: frpoins3xp3g 8182