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Theorem fnwe2val 42464
Description: Lemma for fnwe2 42468. 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 3474 . 2 𝑎 ∈ V
2 vex 3474 . 2 𝑏 ∈ V
3 fveq2 6892 . . . 4 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
4 fveq2 6892 . . . 4 (𝑦 = 𝑏 → (𝐹𝑦) = (𝐹𝑏))
53, 4breqan12d 5159 . . 3 ((𝑥 = 𝑎𝑦 = 𝑏) → ((𝐹𝑥)𝑅(𝐹𝑦) ↔ (𝐹𝑎)𝑅(𝐹𝑏)))
63, 4eqeqan12d 2742 . . . 4 ((𝑥 = 𝑎𝑦 = 𝑏) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑎) = (𝐹𝑏)))
7 simpl 482 . . . . 5 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑥 = 𝑎)
8 fvex 6905 . . . . . . . 8 (𝐹𝑥) ∈ V
9 fnwe2.su . . . . . . . 8 (𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)
108, 9csbie 3926 . . . . . . 7 (𝐹𝑥) / 𝑧𝑆 = 𝑈
113csbeq1d 3894 . . . . . . 7 (𝑥 = 𝑎(𝐹𝑥) / 𝑧𝑆 = (𝐹𝑎) / 𝑧𝑆)
1210, 11eqtr3id 2782 . . . . . 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 5157 . . . 4 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑥𝑈𝑦𝑎(𝐹𝑎) / 𝑧𝑆𝑏))
166, 15anbi12d 631 . . 3 ((𝑥 = 𝑎𝑦 = 𝑏) → (((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦) ↔ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))
175, 16orbi12d 917 . 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 5534 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 846   = wceq 1534  csb 3890   class class class wbr 5143  {copab 5205  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-iota 6495  df-fv 6551
This theorem is referenced by:  fnwe2lem2  42466  fnwe2lem3  42467
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