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Theorem cdleme31fv2 39777
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 23-Feb-2013.)
Hypothesis
Ref Expression
cdleme31fv2.f 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))
Assertion
Ref Expression
cdleme31fv2 ((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝐹𝑋) = 𝑋)
Distinct variable groups:   𝑥,𝐵   𝑥,   𝑥,𝑃   𝑥,𝑄   𝑥,𝑊   𝑥,𝑋
Allowed substitution hints:   𝐹(𝑥)   𝑂(𝑥)

Proof of Theorem cdleme31fv2
StepHypRef Expression
1 cdleme31fv2.f . 2 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))
2 breq1 5144 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥 𝑊𝑋 𝑊))
32notbid 318 . . . . . . . 8 (𝑥 = 𝑋 → (¬ 𝑥 𝑊 ↔ ¬ 𝑋 𝑊))
43anbi2d 628 . . . . . . 7 (𝑥 = 𝑋 → ((𝑃𝑄 ∧ ¬ 𝑥 𝑊) ↔ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)))
54notbid 318 . . . . . 6 (𝑥 = 𝑋 → (¬ (𝑃𝑄 ∧ ¬ 𝑥 𝑊) ↔ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)))
65biimparc 479 . . . . 5 ((¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ 𝑥 = 𝑋) → ¬ (𝑃𝑄 ∧ ¬ 𝑥 𝑊))
76adantll 711 . . . 4 (((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ 𝑥 = 𝑋) → ¬ (𝑃𝑄 ∧ ¬ 𝑥 𝑊))
87iffalsed 4534 . . 3 (((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ 𝑥 = 𝑋) → if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥) = 𝑥)
9 simpr 484 . . 3 (((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
108, 9eqtrd 2766 . 2 (((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ 𝑥 = 𝑋) → if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥) = 𝑋)
11 simpl 482 . 2 ((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → 𝑋𝐵)
121, 10, 11, 11fvmptd2 7000 1 ((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝐹𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1533  wcel 2098  wne 2934  ifcif 4523   class class class wbr 5141  cmpt 5224  cfv 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545
This theorem is referenced by:  cdleme31id  39778  cdleme32fvcl  39824  cdleme32e  39829  cdleme32le  39831  cdleme48gfv  39921  cdleme50ldil  39932
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