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Theorem cdleme31so 37555
 Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 25-Feb-2013.)
Hypotheses
Ref Expression
cdleme31so.o 𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))
cdleme31so.c 𝐶 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊))))
Assertion
Ref Expression
cdleme31so (𝑋𝐵𝑋 / 𝑥𝑂 = 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,   𝑥,   𝑥,   𝑥,𝑁   𝑥,𝑠,𝑧,𝑋   𝑥,𝑊
Allowed substitution hints:   𝐴(𝑧,𝑠)   𝐵(𝑧,𝑠)   𝐶(𝑥,𝑧,𝑠)   (𝑧,𝑠)   (𝑧,𝑠)   (𝑧,𝑠)   𝑁(𝑧,𝑠)   𝑂(𝑥,𝑧,𝑠)   𝑊(𝑧,𝑠)

Proof of Theorem cdleme31so
StepHypRef Expression
1 nfcvd 2975 . . 3 (𝑋𝐵𝑥(𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊)))))
2 oveq1 7137 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥 𝑊) = (𝑋 𝑊))
32oveq2d 7146 . . . . . . . 8 (𝑥 = 𝑋 → (𝑠 (𝑥 𝑊)) = (𝑠 (𝑋 𝑊)))
4 id 22 . . . . . . . 8 (𝑥 = 𝑋𝑥 = 𝑋)
53, 4eqeq12d 2837 . . . . . . 7 (𝑥 = 𝑋 → ((𝑠 (𝑥 𝑊)) = 𝑥 ↔ (𝑠 (𝑋 𝑊)) = 𝑋))
65anbi2d 631 . . . . . 6 (𝑥 = 𝑋 → ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) ↔ (¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋)))
72oveq2d 7146 . . . . . . 7 (𝑥 = 𝑋 → (𝑁 (𝑥 𝑊)) = (𝑁 (𝑋 𝑊)))
87eqeq2d 2832 . . . . . 6 (𝑥 = 𝑋 → (𝑧 = (𝑁 (𝑥 𝑊)) ↔ 𝑧 = (𝑁 (𝑋 𝑊))))
96, 8imbi12d 348 . . . . 5 (𝑥 = 𝑋 → (((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))) ↔ ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊)))))
109ralbidv 3185 . . . 4 (𝑥 = 𝑋 → (∀𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))) ↔ ∀𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊)))))
1110riotabidv 7090 . . 3 (𝑥 = 𝑋 → (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊)))) = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊)))))
121, 11csbiegf 3890 . 2 (𝑋𝐵𝑋 / 𝑥(𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊)))) = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊)))))
13 cdleme31so.o . . 3 𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))
1413csbeq2i 3865 . 2 𝑋 / 𝑥𝑂 = 𝑋 / 𝑥(𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))
15 cdleme31so.c . 2 𝐶 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) → 𝑧 = (𝑁 (𝑋 𝑊))))
1612, 14, 153eqtr4g 2881 1 (𝑋𝐵𝑋 / 𝑥𝑂 = 𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∀wral 3126  ⦋csb 3857   class class class wbr 5039  ℩crio 7087  (class class class)co 7130 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-iota 6287  df-fv 6336  df-riota 7088  df-ov 7133 This theorem is referenced by:  cdleme31fv1s  37568  cdlemefrs32fva  37576  cdleme32fva  37613
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