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Theorem cdleme31sn1c 35993
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 2651 . . 3 (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺)) = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺))
41, 2, 3cdleme31sn1 35986 . 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 35987 . . . . . . . 8 (𝑅𝐴𝑅 / 𝑠𝐺 = 𝑌)
87adantr 480 . . . . . . 7 ((𝑅𝐴𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝐺 = 𝑌)
98eqeq2d 2661 . . . . . 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 3015 . . . 4 ((𝑅𝐴𝑅 (𝑃 𝑄)) → (∀𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺) ↔ ∀𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑌)))
1211riotabidv 6653 . . 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 2703 . 2 ((𝑅𝐴𝑅 (𝑃 𝑄)) → (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺)) = 𝐶)
154, 14eqtrd 2685 1 ((𝑅𝐴𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  csb 3566  ifcif 4119   class class class wbr 4685  crio 6650  (class class class)co 6690
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-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-riota 6651  df-ov 6693
This theorem is referenced by:  cdlemefs32sn1aw  36019  cdleme43fsv1snlem  36025  cdleme41sn3a  36038  cdleme40m  36072  cdleme40n  36073
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