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Theorem cdleme31fv2 39727
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 5151 . . . . . . . . 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 4539 . . 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 2771 . 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 7006 1 ((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝐹𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2105  wne 2939  ifcif 4528   class class class wbr 5148  cmpt 5231  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551
This theorem is referenced by:  cdleme31id  39728  cdleme32fvcl  39774  cdleme32e  39779  cdleme32le  39781  cdleme48gfv  39871  cdleme50ldil  39882
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