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Theorem ralxp3es 8148

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 3796	. 2 ⊢ Ⅎ𝑦[(1^st ‘(1^st ‘𝑥)) / 𝑦][(2^nd ‘(1^st ‘𝑥)) / 𝑧][(2^nd ‘𝑥) / 𝑤]𝜑
2		nfcv 2898	. . 3 ⊢ Ⅎ𝑧(1^st ‘(1^st ‘𝑥))
3		nfsbc1v 3796	. . 3 ⊢ Ⅎ𝑧[(2^nd ‘(1^st ‘𝑥)) / 𝑧][(2^nd ‘𝑥) / 𝑤]𝜑
4	2, 3	nfsbcw 3798	. 2 ⊢ Ⅎ𝑧[(1^st ‘(1^st ‘𝑥)) / 𝑦][(2^nd ‘(1^st ‘𝑥)) / 𝑧][(2^nd ‘𝑥) / 𝑤]𝜑
5		nfcv 2898	. . 3 ⊢ Ⅎ𝑤(1^st ‘(1^st ‘𝑥))
6		nfcv 2898	. . . 4 ⊢ Ⅎ𝑤(2^nd ‘(1^st ‘𝑥))
7		nfsbc1v 3796	. . . 4 ⊢ Ⅎ𝑤[(2^nd ‘𝑥) / 𝑤]𝜑
8	6, 7	nfsbcw 3798	. . 3 ⊢ Ⅎ𝑤[(2^nd ‘(1^st ‘𝑥)) / 𝑧][(2^nd ‘𝑥) / 𝑤]𝜑
9	5, 8	nfsbcw 3798	. 2 ⊢ Ⅎ𝑤[(1^st ‘(1^st ‘𝑥)) / 𝑦][(2^nd ‘(1^st ‘𝑥)) / 𝑧][(2^nd ‘𝑥) / 𝑤]𝜑
10		nfv 1909	. 2 ⊢ Ⅎ𝑥𝜑
11		sbcoteq1a 8059	. 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 8146	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 205 ∀wral 3057 [wsbc 3776 × cxp 5678 ‘cfv 6551 1^st c1st 7995 2^nd c2nd 7996

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 ax-un 7744

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-ot 4639 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-iota 6503 df-fun 6553 df-fv 6559 df-1st 7997 df-2nd 7998

This theorem is referenced by: frpoins3xp3g 8150