Theorem ralxpes 8133

Description: A version of ralxp 5829 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 3765	. 2 ⊢ Ⅎ𝑦[(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑
2		nfcv 2925	. . 3 ⊢ Ⅎ𝑧(1st ‘𝑥)
3		nfsbc1v 3765	. . 3 ⊢ Ⅎ𝑧[(2nd ‘𝑥) / 𝑧]𝜑
4	2, 3	nfsbcw 3767	. 2 ⊢ Ⅎ𝑧[(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑
5		nfv 1944	. 2 ⊢ Ⅎ𝑥𝜑
6		sbcopeq1a 8047	. 2 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ([(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑 ↔ 𝜑))
7	1, 4, 5, 6	ralxpf 5834	1 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)[(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜑)

Syntax hints:   ↔ wb 209  ∀wral 3079  [wsbc 3745   × cxp 5661  ‘cfv 6538  1^st c1st 7985  2^nd c2nd 7986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fv 6546  df-1st 7987  df-2nd 7988

This theorem is referenced by:  frpoins3xpg  8137