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Theorem vtocl2d 28505
Description: Implicit substitution of two classes for two setvar variables. (Contributed by Thierry Arnoux, 25-Aug-2020.)
Hypotheses
Ref Expression
vtocl2d.a (𝜑𝐴𝑉)
vtocl2d.b (𝜑𝐵𝑊)
vtocl2d.1 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒))
vtocl2d.3 (𝜑𝜓)
Assertion
Ref Expression
vtocl2d (𝜑𝜒)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑉   𝑥,𝑊,𝑦   𝜒,𝑥,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝑉(𝑦)

Proof of Theorem vtocl2d
StepHypRef Expression
1 vtocl2d.b . . 3 (𝜑𝐵𝑊)
2 vtocl2d.a . . 3 (𝜑𝐴𝑉)
3 nfcv 2750 . . . 4 𝑦𝐵
4 nfcv 2750 . . . 4 𝑥𝐵
5 nfcv 2750 . . . 4 𝑥𝐴
6 nfv 1829 . . . . 5 𝑦𝜑
7 nfsbc1v 3421 . . . . 5 𝑦[𝐵 / 𝑦]𝜓
86, 7nfim 1812 . . . 4 𝑦(𝜑[𝐵 / 𝑦]𝜓)
9 nfv 1829 . . . 4 𝑥(𝜑𝜒)
10 sbceq1a 3412 . . . . 5 (𝑦 = 𝐵 → (𝜓[𝐵 / 𝑦]𝜓))
1110imbi2d 328 . . . 4 (𝑦 = 𝐵 → ((𝜑𝜓) ↔ (𝜑[𝐵 / 𝑦]𝜓)))
12 sbceq1a 3412 . . . . . . 7 (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜓[𝐴 / 𝑥][𝐵 / 𝑦]𝜓))
1312adantr 479 . . . . . 6 ((𝑥 = 𝐴𝜑) → ([𝐵 / 𝑦]𝜓[𝐴 / 𝑥][𝐵 / 𝑦]𝜓))
14 nfv 1829 . . . . . . . . 9 𝑥𝜒
15 nfv 1829 . . . . . . . . 9 𝑦𝜒
16 nfv 1829 . . . . . . . . 9 𝑥 𝐵𝑊
17 vtocl2d.1 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒))
1814, 15, 16, 17sbc2iegf 3470 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓𝜒))
192, 1, 18syl2anc 690 . . . . . . 7 (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓𝜒))
2019adantl 480 . . . . . 6 ((𝑥 = 𝐴𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓𝜒))
2113, 20bitrd 266 . . . . 5 ((𝑥 = 𝐴𝜑) → ([𝐵 / 𝑦]𝜓𝜒))
2221pm5.74da 718 . . . 4 (𝑥 = 𝐴 → ((𝜑[𝐵 / 𝑦]𝜓) ↔ (𝜑𝜒)))
23 vtocl2d.3 . . . 4 (𝜑𝜓)
243, 4, 5, 8, 9, 11, 22, 23vtocl2gf 3240 . . 3 ((𝐵𝑊𝐴𝑉) → (𝜑𝜒))
251, 2, 24syl2anc 690 . 2 (𝜑 → (𝜑𝜒))
2625pm2.43i 49 1 (𝜑𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  [wsbc 3401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-v 3174  df-sbc 3402
This theorem is referenced by:  submateq  29009
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