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Theorem ralxp3es 8093

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 3762	. 2 ⊢ Ⅎ𝑦[(1^st ‘(1^st ‘𝑥)) / 𝑦][(2^nd ‘(1^st ‘𝑥)) / 𝑧][(2^nd ‘𝑥) / 𝑤]𝜑
2		nfcv 2899	. . 3 ⊢ Ⅎ𝑧(1^st ‘(1^st ‘𝑥))
3		nfsbc1v 3762	. . 3 ⊢ Ⅎ𝑧[(2^nd ‘(1^st ‘𝑥)) / 𝑧][(2^nd ‘𝑥) / 𝑤]𝜑
4	2, 3	nfsbcw 3764	. 2 ⊢ Ⅎ𝑧[(1^st ‘(1^st ‘𝑥)) / 𝑦][(2^nd ‘(1^st ‘𝑥)) / 𝑧][(2^nd ‘𝑥) / 𝑤]𝜑
5		nfcv 2899	. . 3 ⊢ Ⅎ𝑤(1^st ‘(1^st ‘𝑥))
6		nfcv 2899	. . . 4 ⊢ Ⅎ𝑤(2^nd ‘(1^st ‘𝑥))
7		nfsbc1v 3762	. . . 4 ⊢ Ⅎ𝑤[(2^nd ‘𝑥) / 𝑤]𝜑
8	6, 7	nfsbcw 3764	. . 3 ⊢ Ⅎ𝑤[(2^nd ‘(1^st ‘𝑥)) / 𝑧][(2^nd ‘𝑥) / 𝑤]𝜑
9	5, 8	nfsbcw 3764	. 2 ⊢ Ⅎ𝑤[(1^st ‘(1^st ‘𝑥)) / 𝑦][(2^nd ‘(1^st ‘𝑥)) / 𝑧][(2^nd ‘𝑥) / 𝑤]𝜑
10		nfv 1916	. 2 ⊢ Ⅎ𝑥𝜑
11		sbcoteq1a 8007	. 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 8091	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 3052 [wsbc 3742 × cxp 5632 ‘cfv 6502 1^st c1st 7943 2^nd c2nd 7944

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5245 ax-nul 5255 ax-pr 5381 ax-un 7692

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-sn 4583 df-pr 4585 df-op 4589 df-ot 4591 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5529 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6458 df-fun 6504 df-fv 6510 df-1st 7945 df-2nd 7946

This theorem is referenced by: frpoins3xp3g 8095