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Theorem fnwe2val 39800
Description: Lemma for fnwe2 39804. 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 6643 . . . 4 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
4 fveq2 6643 . . . 4 (𝑦 = 𝑏 → (𝐹𝑦) = (𝐹𝑏))
53, 4breqan12d 5055 . . 3 ((𝑥 = 𝑎𝑦 = 𝑏) → ((𝐹𝑥)𝑅(𝐹𝑦) ↔ (𝐹𝑎)𝑅(𝐹𝑏)))
63, 4eqeqan12d 2838 . . . 4 ((𝑥 = 𝑎𝑦 = 𝑏) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑎) = (𝐹𝑏)))
7 simpl 486 . . . . 5 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑥 = 𝑎)
8 fvex 6656 . . . . . . . 8 (𝐹𝑥) ∈ V
9 fnwe2.su . . . . . . . 8 (𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)
108, 9csbie 3892 . . . . . . 7 (𝐹𝑥) / 𝑧𝑆 = 𝑈
113csbeq1d 3861 . . . . . . 7 (𝑥 = 𝑎(𝐹𝑥) / 𝑧𝑆 = (𝐹𝑎) / 𝑧𝑆)
1210, 11syl5eqr 2870 . . . . . 6 (𝑥 = 𝑎𝑈 = (𝐹𝑎) / 𝑧𝑆)
1312adantr 484 . . . . 5 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑈 = (𝐹𝑎) / 𝑧𝑆)
14 simpr 488 . . . . 5 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑦 = 𝑏)
157, 13, 14breq123d 5053 . . . 4 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑥𝑈𝑦𝑎(𝐹𝑎) / 𝑧𝑆𝑏))
166, 15anbi12d 633 . . 3 ((𝑥 = 𝑎𝑦 = 𝑏) → (((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦) ↔ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))
175, 16orbi12d 916 . 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 5397 1 (𝑎𝑇𝑏 ↔ ((𝐹𝑎)𝑅(𝐹𝑏) ∨ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844   = wceq 1538  csb 3857   class class class wbr 5039  {copab 5101  cfv 6328
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  ax-sep 5176  ax-nul 5183  ax-pr 5303
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-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  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-opab 5102  df-iota 6287  df-fv 6336
This theorem is referenced by:  fnwe2lem2  39802  fnwe2lem3  39803
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