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Theorem cdleme31sn1c 40370
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 2734 . . 3 (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺)) = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺))
41, 2, 3cdleme31sn1 40363 . 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 40364 . . . . . . . 8 (𝑅𝐴𝑅 / 𝑠𝐺 = 𝑌)
87adantr 480 . . . . . . 7 ((𝑅𝐴𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝐺 = 𝑌)
98eqeq2d 2745 . . . . . 6 ((𝑅𝐴𝑅 (𝑃 𝑄)) → (𝑦 = 𝑅 / 𝑠𝐺𝑦 = 𝑌))
109imbi2d 340 . . . . 5 ((𝑅𝐴𝑅 (𝑃 𝑄)) → (((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺) ↔ ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑌)))
1110ralbidv 3175 . . . 4 ((𝑅𝐴𝑅 (𝑃 𝑄)) → (∀𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺) ↔ ∀𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑌)))
1211riotabidv 7389 . . 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, 13eqtr4di 2792 . 2 ((𝑅𝐴𝑅 (𝑃 𝑄)) → (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺)) = 𝐶)
154, 14eqtrd 2774 1 ((𝑅𝐴𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1536  wcel 2105  wral 3058  csb 3907  ifcif 4530   class class class wbr 5147  crio 7386  (class class class)co 7430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-riota 7387  df-ov 7433
This theorem is referenced by:  cdlemefs32sn1aw  40396  cdleme43fsv1snlem  40402  cdleme41sn3a  40415  cdleme40m  40449  cdleme40n  40450
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