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Theorem cdleme31fv2 38052
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 5033 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥 𝑊𝑋 𝑊))
32notbid 321 . . . . . . . 8 (𝑥 = 𝑋 → (¬ 𝑥 𝑊 ↔ ¬ 𝑋 𝑊))
43anbi2d 632 . . . . . . 7 (𝑥 = 𝑋 → ((𝑃𝑄 ∧ ¬ 𝑥 𝑊) ↔ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)))
54notbid 321 . . . . . 6 (𝑥 = 𝑋 → (¬ (𝑃𝑄 ∧ ¬ 𝑥 𝑊) ↔ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)))
65biimparc 483 . . . . 5 ((¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ 𝑥 = 𝑋) → ¬ (𝑃𝑄 ∧ ¬ 𝑥 𝑊))
76adantll 714 . . . 4 (((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ 𝑥 = 𝑋) → ¬ (𝑃𝑄 ∧ ¬ 𝑥 𝑊))
87iffalsed 4425 . . 3 (((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ 𝑥 = 𝑋) → if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥) = 𝑥)
9 simpr 488 . . 3 (((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
108, 9eqtrd 2773 . 2 (((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ 𝑥 = 𝑋) → if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥) = 𝑋)
11 simpl 486 . 2 ((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → 𝑋𝐵)
121, 10, 11, 11fvmptd2 6785 1 ((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝐹𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1542  wcel 2114  wne 2934  ifcif 4414   class class class wbr 5030  cmpt 5110  cfv 6339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6297  df-fun 6341  df-fv 6347
This theorem is referenced by:  cdleme31id  38053  cdleme32fvcl  38099  cdleme32e  38104  cdleme32le  38106  cdleme48gfv  38196  cdleme50ldil  38207
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