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Theorem csbxpgOLD 39368
Description: Distribute proper substitution through the Cartesian product of two classes. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 23-Aug-2018. Use csbrn 5631 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
csbxpgOLD (𝐴𝐷𝐴 / 𝑥(𝐵 × 𝐶) = (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶))

Proof of Theorem csbxpgOLD
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbab 4041 . . 3 𝐴 / 𝑥{𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} = {𝑧[𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))}
2 sbcex2 3519 . . . . 5 ([𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤[𝐴 / 𝑥]𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)))
3 sbcex2 3519 . . . . . . 7 ([𝐴 / 𝑥]𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)))
4 sbcan 3511 . . . . . . . . 9 ([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤𝐵𝑦𝐶)))
5 sbcg 3536 . . . . . . . . . 10 (𝐴𝐷 → ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ↔ 𝑧 = ⟨𝑤, 𝑦⟩))
6 sbcan 3511 . . . . . . . . . . 11 ([𝐴 / 𝑥](𝑤𝐵𝑦𝐶) ↔ ([𝐴 / 𝑥]𝑤𝐵[𝐴 / 𝑥]𝑦𝐶))
7 sbcel2 4022 . . . . . . . . . . . . 13 ([𝐴 / 𝑥]𝑤𝐵𝑤𝐴 / 𝑥𝐵)
87a1i 11 . . . . . . . . . . . 12 (𝐴𝐷 → ([𝐴 / 𝑥]𝑤𝐵𝑤𝐴 / 𝑥𝐵))
9 sbcel2 4022 . . . . . . . . . . . . 13 ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶)
109a1i 11 . . . . . . . . . . . 12 (𝐴𝐷 → ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶))
118, 10anbi12d 747 . . . . . . . . . . 11 (𝐴𝐷 → (([𝐴 / 𝑥]𝑤𝐵[𝐴 / 𝑥]𝑦𝐶) ↔ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))
126, 11syl5bb 272 . . . . . . . . . 10 (𝐴𝐷 → ([𝐴 / 𝑥](𝑤𝐵𝑦𝐶) ↔ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))
135, 12anbi12d 747 . . . . . . . . 9 (𝐴𝐷 → (([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤𝐵𝑦𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))))
144, 13syl5bb 272 . . . . . . . 8 (𝐴𝐷 → ([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))))
1514exbidv 1890 . . . . . . 7 (𝐴𝐷 → (∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))))
163, 15syl5bb 272 . . . . . 6 (𝐴𝐷 → ([𝐴 / 𝑥]𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))))
1716exbidv 1890 . . . . 5 (𝐴𝐷 → (∃𝑤[𝐴 / 𝑥]𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))))
182, 17syl5bb 272 . . . 4 (𝐴𝐷 → ([𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))))
1918abbidv 2770 . . 3 (𝐴𝐷 → {𝑧[𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))})
201, 19syl5eq 2697 . 2 (𝐴𝐷𝐴 / 𝑥{𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))})
21 df-xp 5149 . . . 4 (𝐵 × 𝐶) = {⟨𝑤, 𝑦⟩ ∣ (𝑤𝐵𝑦𝐶)}
22 df-opab 4746 . . . 4 {⟨𝑤, 𝑦⟩ ∣ (𝑤𝐵𝑦𝐶)} = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))}
2321, 22eqtri 2673 . . 3 (𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))}
2423csbeq2i 4026 . 2 𝐴 / 𝑥(𝐵 × 𝐶) = 𝐴 / 𝑥{𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))}
25 df-xp 5149 . . 3 (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶) = {⟨𝑤, 𝑦⟩ ∣ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)}
26 df-opab 4746 . . 3 {⟨𝑤, 𝑦⟩ ∣ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)} = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))}
2725, 26eqtri 2673 . 2 (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶) = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))}
2820, 24, 273eqtr4g 2710 1 (𝐴𝐷𝐴 / 𝑥(𝐵 × 𝐶) = (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  {cab 2637  [wsbc 3468  csb 3566  cop 4216  {copab 4745   × cxp 5141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-nul 3949  df-opab 4746  df-xp 5149
This theorem is referenced by:  csbresgVD  39445
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