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Theorem cdleme31fv2 34499
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 4577 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥 𝑊𝑋 𝑊))
43notbid 306 . . . . . . . 8 (𝑥 = 𝑋 → (¬ 𝑥 𝑊 ↔ ¬ 𝑋 𝑊))
54anbi2d 735 . . . . . . 7 (𝑥 = 𝑋 → ((𝑃𝑄 ∧ ¬ 𝑥 𝑊) ↔ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)))
65notbid 306 . . . . . 6 (𝑥 = 𝑋 → (¬ (𝑃𝑄 ∧ ¬ 𝑥 𝑊) ↔ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)))
76biimparc 502 . . . . 5 ((¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ 𝑥 = 𝑋) → ¬ (𝑃𝑄 ∧ ¬ 𝑥 𝑊))
87adantll 745 . . . 4 (((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ 𝑥 = 𝑋) → ¬ (𝑃𝑄 ∧ ¬ 𝑥 𝑊))
98iffalsed 4043 . . 3 (((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ 𝑥 = 𝑋) → if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥) = 𝑥)
10 simpr 475 . . 3 (((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
119, 10eqtrd 2640 . 2 (((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ 𝑥 = 𝑋) → if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥) = 𝑋)
12 simpl 471 . 2 ((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → 𝑋𝐵)
132, 11, 12, 12fvmptd 6179 1 ((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝐹𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wcel 1976  wne 2776  ifcif 4032   class class class wbr 4574  cmpt 4634  cfv 5787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pr 4825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-iota 5751  df-fun 5789  df-fv 5795
This theorem is referenced by:  cdleme31id  34500  cdleme32fvcl  34546  cdleme32e  34551  cdleme32le  34553  cdleme48gfv  34643  cdleme50ldil  34654
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