Theorem ralxpes 8177

Description: A version of ralxp 5866 with explicit substitution. (Contributed by Scott Fenton, 21-Aug-2024.)

Assertion

Ref	Expression
ralxpes	⊢ (∀𝑥 ∈ (𝐴 × 𝐵)[(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜑)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦,𝑧   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐴(𝑧)

Proof of Theorem ralxpes

Step	Hyp	Ref	Expression
1		nfsbc1v 3824	. 2 ⊢ Ⅎ𝑦[(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑
2		nfcv 2908	. . 3 ⊢ Ⅎ𝑧(1st ‘𝑥)
3		nfsbc1v 3824	. . 3 ⊢ Ⅎ𝑧[(2nd ‘𝑥) / 𝑧]𝜑
4	2, 3	nfsbcw 3826	. 2 ⊢ Ⅎ𝑧[(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑
5		nfv 1913	. 2 ⊢ Ⅎ𝑥𝜑
6		sbcopeq1a 8090	. 2 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ([(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑 ↔ 𝜑))
7	1, 4, 5, 6	ralxpf 5871	1 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)[(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜑)

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-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:  frpoins3xpg  8181