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Theorem cdleme31fv2 36414
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 . . 3 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))
21a1i 11 . 2 ((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥)))
3 breq1 4846 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥 𝑊𝑋 𝑊))
43notbid 310 . . . . . . . 8 (𝑥 = 𝑋 → (¬ 𝑥 𝑊 ↔ ¬ 𝑋 𝑊))
54anbi2d 623 . . . . . . 7 (𝑥 = 𝑋 → ((𝑃𝑄 ∧ ¬ 𝑥 𝑊) ↔ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)))
65notbid 310 . . . . . 6 (𝑥 = 𝑋 → (¬ (𝑃𝑄 ∧ ¬ 𝑥 𝑊) ↔ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)))
76biimparc 472 . . . . 5 ((¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ 𝑥 = 𝑋) → ¬ (𝑃𝑄 ∧ ¬ 𝑥 𝑊))
87adantll 706 . . . 4 (((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ 𝑥 = 𝑋) → ¬ (𝑃𝑄 ∧ ¬ 𝑥 𝑊))
98iffalsed 4288 . . 3 (((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ 𝑥 = 𝑋) → if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥) = 𝑥)
10 simpr 478 . . 3 (((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
119, 10eqtrd 2833 . 2 (((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ 𝑥 = 𝑋) → if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥) = 𝑋)
12 simpl 475 . 2 ((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → 𝑋𝐵)
132, 11, 12, 12fvmptd 6513 1 ((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝐹𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385   = wceq 1653  wcel 2157  wne 2971  ifcif 4277   class class class wbr 4843  cmpt 4922  cfv 6101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-iota 6064  df-fun 6103  df-fv 6109
This theorem is referenced by:  cdleme31id  36415  cdleme32fvcl  36461  cdleme32e  36466  cdleme32le  36468  cdleme48gfv  36558  cdleme50ldil  36569
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