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Theorem fnwe2val 42366
Description: Lemma for fnwe2 42370. Substitute variables. (Contributed by Stefan O'Rear, 19-Jan-2015.)
Hypotheses
Ref Expression
fnwe2.su (𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)
fnwe2.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦))}
Assertion
Ref Expression
fnwe2val (𝑎𝑇𝑏 ↔ ((𝐹𝑎)𝑅(𝐹𝑏) ∨ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))
Distinct variable groups:   𝑦,𝑈,𝑧,𝑎,𝑏   𝑥,𝑆,𝑦,𝑎,𝑏   𝑥,𝑅,𝑦,𝑎,𝑏   𝑥,𝑧,𝐹,𝑦,𝑎,𝑏   𝑇,𝑎,𝑏
Allowed substitution hints:   𝑅(𝑧)   𝑆(𝑧)   𝑇(𝑥,𝑦,𝑧)   𝑈(𝑥)

Proof of Theorem fnwe2val
StepHypRef Expression
1 vex 3472 . 2 𝑎 ∈ V
2 vex 3472 . 2 𝑏 ∈ V
3 fveq2 6885 . . . 4 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
4 fveq2 6885 . . . 4 (𝑦 = 𝑏 → (𝐹𝑦) = (𝐹𝑏))
53, 4breqan12d 5157 . . 3 ((𝑥 = 𝑎𝑦 = 𝑏) → ((𝐹𝑥)𝑅(𝐹𝑦) ↔ (𝐹𝑎)𝑅(𝐹𝑏)))
63, 4eqeqan12d 2740 . . . 4 ((𝑥 = 𝑎𝑦 = 𝑏) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑎) = (𝐹𝑏)))
7 simpl 482 . . . . 5 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑥 = 𝑎)
8 fvex 6898 . . . . . . . 8 (𝐹𝑥) ∈ V
9 fnwe2.su . . . . . . . 8 (𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)
108, 9csbie 3924 . . . . . . 7 (𝐹𝑥) / 𝑧𝑆 = 𝑈
113csbeq1d 3892 . . . . . . 7 (𝑥 = 𝑎(𝐹𝑥) / 𝑧𝑆 = (𝐹𝑎) / 𝑧𝑆)
1210, 11eqtr3id 2780 . . . . . 6 (𝑥 = 𝑎𝑈 = (𝐹𝑎) / 𝑧𝑆)
1312adantr 480 . . . . 5 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑈 = (𝐹𝑎) / 𝑧𝑆)
14 simpr 484 . . . . 5 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑦 = 𝑏)
157, 13, 14breq123d 5155 . . . 4 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑥𝑈𝑦𝑎(𝐹𝑎) / 𝑧𝑆𝑏))
166, 15anbi12d 630 . . 3 ((𝑥 = 𝑎𝑦 = 𝑏) → (((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦) ↔ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))
175, 16orbi12d 915 . 2 ((𝑥 = 𝑎𝑦 = 𝑏) → (((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦)) ↔ ((𝐹𝑎)𝑅(𝐹𝑏) ∨ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏))))
18 fnwe2.t . 2 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦))}
191, 2, 17, 18braba 5530 1 (𝑎𝑇𝑏 ↔ ((𝐹𝑎)𝑅(𝐹𝑏) ∨ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wo 844   = wceq 1533  csb 3888   class class class wbr 5141  {copab 5203  cfv 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-iota 6489  df-fv 6545
This theorem is referenced by:  fnwe2lem2  42368  fnwe2lem3  42369
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