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Theorem cdleme31sn1c 34694
 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 2610 . . 3 (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺)) = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺))
41, 2, 3cdleme31sn1 34687 . 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 34688 . . . . . . . 8 (𝑅𝐴𝑅 / 𝑠𝐺 = 𝑌)
87adantr 480 . . . . . . 7 ((𝑅𝐴𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝐺 = 𝑌)
98eqeq2d 2620 . . . . . 6 ((𝑅𝐴𝑅 (𝑃 𝑄)) → (𝑦 = 𝑅 / 𝑠𝐺𝑦 = 𝑌))
109imbi2d 329 . . . . 5 ((𝑅𝐴𝑅 (𝑃 𝑄)) → (((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺) ↔ ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑌)))
1110ralbidv 2969 . . . 4 ((𝑅𝐴𝑅 (𝑃 𝑄)) → (∀𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺) ↔ ∀𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑌)))
1211riotabidv 6513 . . 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 2662 . 2 ((𝑅𝐴𝑅 (𝑃 𝑄)) → (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺)) = 𝐶)
154, 14eqtrd 2644 1 ((𝑅𝐴𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁 = 𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ⦋csb 3499  ifcif 4036   class class class wbr 4583  ℩crio 6510  (class class class)co 6549 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-riota 6511  df-ov 6552 This theorem is referenced by:  cdlemefs32sn1aw  34720  cdleme43fsv1snlem  34726  cdleme41sn3a  34739  cdleme40m  34773  cdleme40n  34774
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