Theorem ralxpes 8118

Description: A version of ralxp 5808 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 3776	. 2 ⊢ Ⅎ𝑦[(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑
2		nfcv 2892	. . 3 ⊢ Ⅎ𝑧(1st ‘𝑥)
3		nfsbc1v 3776	. . 3 ⊢ Ⅎ𝑧[(2nd ‘𝑥) / 𝑧]𝜑
4	2, 3	nfsbcw 3778	. 2 ⊢ Ⅎ𝑧[(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑
5		nfv 1914	. 2 ⊢ Ⅎ𝑥𝜑
6		sbcopeq1a 8031	. 2 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ([(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑 ↔ 𝜑))
7	1, 4, 5, 6	ralxpf 5813	1 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)[(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜑)

Syntax hints:   ↔ wb 206  ∀wral 3045  [wsbc 3756   × cxp 5639  ‘cfv 6514  1^st c1st 7969  2^nd c2nd 7970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fv 6522  df-1st 7971  df-2nd 7972

This theorem is referenced by:  frpoins3xpg  8122