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Mirrors > Home > MPE Home > Th. List > Mathboxes > ralxpes | Structured version Visualization version GIF version |
Description: A version of ralxp 5687 with explicit substitution. (Contributed by Scott Fenton, 21-Aug-2024.) |
Ref | Expression |
---|---|
ralxpes | ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)[(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfsbc1v 3718 | . 2 ⊢ Ⅎ𝑦[(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑 | |
2 | nfcv 2919 | . . 3 ⊢ Ⅎ𝑧(1st ‘𝑥) | |
3 | nfsbc1v 3718 | . . 3 ⊢ Ⅎ𝑧[(2nd ‘𝑥) / 𝑧]𝜑 | |
4 | 2, 3 | nfsbcw 3720 | . 2 ⊢ Ⅎ𝑧[(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑 |
5 | nfv 1915 | . 2 ⊢ Ⅎ𝑥𝜑 | |
6 | sbcopeq1a 7758 | . 2 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → ([(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑 ↔ 𝜑)) | |
7 | 1, 4, 5, 6 | ralxpf 5692 | 1 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)[(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∀wral 3070 [wsbc 3698 × cxp 5526 ‘cfv 6340 1st c1st 7697 2nd c2nd 7698 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-sep 5173 ax-nul 5180 ax-pr 5302 ax-un 7465 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-id 5434 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-iota 6299 df-fun 6342 df-fv 6348 df-1st 7699 df-2nd 7700 |
This theorem is referenced by: frpoins3xpg 33345 |
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