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Theorem cdleme31sn1c 34493
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 1-Mar-2013.)
Hypotheses
Ref Expression
cdleme31sn1c.g 𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))
cdleme31sn1c.i 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))
cdleme31sn1c.n 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)
cdleme31sn1c.y 𝑌 = ((𝑃 𝑄) (𝐸 ((𝑅 𝑡) 𝑊)))
cdleme31sn1c.c 𝐶 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑌))
Assertion
Ref Expression
cdleme31sn1c ((𝑅𝐴𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁 = 𝐶)
Distinct variable groups:   𝑡,𝑠,𝑦,𝐴   𝐵,𝑠   𝐸,𝑠   ,𝑠,𝑡,𝑦   ,𝑠,𝑡,𝑦   ,𝑠   𝑃,𝑠,𝑡,𝑦   𝑄,𝑠,𝑡,𝑦   𝑅,𝑠,𝑡,𝑦   𝑊,𝑠
Allowed substitution hints:   𝐵(𝑦,𝑡)   𝐶(𝑦,𝑡,𝑠)   𝐷(𝑦,𝑡,𝑠)   𝐸(𝑦,𝑡)   𝐺(𝑦,𝑡,𝑠)   𝐼(𝑦,𝑡,𝑠)   (𝑦,𝑡)   𝑁(𝑦,𝑡,𝑠)   𝑊(𝑦,𝑡)   𝑌(𝑦,𝑡,𝑠)

Proof of Theorem cdleme31sn1c
StepHypRef Expression
1 cdleme31sn1c.i . . 3 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))
2 cdleme31sn1c.n . . 3 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)
3 eqid 2605 . . 3 (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺)) = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺))
41, 2, 3cdleme31sn1 34486 . 2 ((𝑅𝐴𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺)))
5 cdleme31sn1c.g . . . . . . . . 9 𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))
6 cdleme31sn1c.y . . . . . . . . 9 𝑌 = ((𝑃 𝑄) (𝐸 ((𝑅 𝑡) 𝑊)))
75, 6cdleme31se 34487 . . . . . . . 8 (𝑅𝐴𝑅 / 𝑠𝐺 = 𝑌)
87adantr 479 . . . . . . 7 ((𝑅𝐴𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝐺 = 𝑌)
98eqeq2d 2615 . . . . . 6 ((𝑅𝐴𝑅 (𝑃 𝑄)) → (𝑦 = 𝑅 / 𝑠𝐺𝑦 = 𝑌))
109imbi2d 328 . . . . 5 ((𝑅𝐴𝑅 (𝑃 𝑄)) → (((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺) ↔ ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑌)))
1110ralbidv 2964 . . . 4 ((𝑅𝐴𝑅 (𝑃 𝑄)) → (∀𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺) ↔ ∀𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑌)))
1211riotabidv 6487 . . 3 ((𝑅𝐴𝑅 (𝑃 𝑄)) → (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺)) = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑌)))
13 cdleme31sn1c.c . . 3 𝐶 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑌))
1412, 13syl6eqr 2657 . 2 ((𝑅𝐴𝑅 (𝑃 𝑄)) → (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺)) = 𝐶)
154, 14eqtrd 2639 1 ((𝑅𝐴𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wcel 1975  wral 2891  csb 3494  ifcif 4031   class class class wbr 4573  crio 6484  (class class class)co 6523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
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 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-iota 5750  df-fv 5794  df-riota 6485  df-ov 6526
This theorem is referenced by:  cdlemefs32sn1aw  34519  cdleme43fsv1snlem  34525  cdleme41sn3a  34538  cdleme40m  34572  cdleme40n  34573
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