Theorem ralxpes 8120

Description: A version of ralxp 5818 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 3767	. 2 ⊢ Ⅎ𝑦[(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑
2		nfcv 2927	. . 3 ⊢ Ⅎ𝑧(1st ‘𝑥)
3		nfsbc1v 3767	. . 3 ⊢ Ⅎ𝑧[(2nd ‘𝑥) / 𝑧]𝜑
4	2, 3	nfsbcw 3769	. 2 ⊢ Ⅎ𝑧[(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑
5		nfv 1937	. 2 ⊢ Ⅎ𝑥𝜑
6		sbcopeq1a 8034	. 2 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ([(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑 ↔ 𝜑))
7	1, 4, 5, 6	ralxpf 5823	1 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)[(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜑)

Syntax hints:   ↔ wb 209  ∀wral 3079  [wsbc 3747   × cxp 5650  ‘cfv 6525  1^st c1st 7972  2^nd c2nd 7973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-iota 6481  df-fun 6527  df-fv 6533  df-1st 7974  df-2nd 7975

This theorem is referenced by:  frpoins3xpg  8124