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Theorem fnwe2val 38457
Description: Lemma for fnwe2 38461. 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 3417 . 2 𝑎 ∈ V
2 vex 3417 . 2 𝑏 ∈ V
3 fveq2 6437 . . . 4 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
4 fveq2 6437 . . . 4 (𝑦 = 𝑏 → (𝐹𝑦) = (𝐹𝑏))
53, 4breqan12d 4891 . . 3 ((𝑥 = 𝑎𝑦 = 𝑏) → ((𝐹𝑥)𝑅(𝐹𝑦) ↔ (𝐹𝑎)𝑅(𝐹𝑏)))
63, 4eqeqan12d 2841 . . . 4 ((𝑥 = 𝑎𝑦 = 𝑏) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑎) = (𝐹𝑏)))
7 simpl 476 . . . . 5 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑥 = 𝑎)
8 fvex 6450 . . . . . . . 8 (𝐹𝑥) ∈ V
9 fnwe2.su . . . . . . . 8 (𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)
108, 9csbie 3783 . . . . . . 7 (𝐹𝑥) / 𝑧𝑆 = 𝑈
113csbeq1d 3764 . . . . . . 7 (𝑥 = 𝑎(𝐹𝑥) / 𝑧𝑆 = (𝐹𝑎) / 𝑧𝑆)
1210, 11syl5eqr 2875 . . . . . 6 (𝑥 = 𝑎𝑈 = (𝐹𝑎) / 𝑧𝑆)
1312adantr 474 . . . . 5 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑈 = (𝐹𝑎) / 𝑧𝑆)
14 simpr 479 . . . . 5 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑦 = 𝑏)
157, 13, 14breq123d 4889 . . . 4 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑥𝑈𝑦𝑎(𝐹𝑎) / 𝑧𝑆𝑏))
166, 15anbi12d 624 . . 3 ((𝑥 = 𝑎𝑦 = 𝑏) → (((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦) ↔ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))
175, 16orbi12d 947 . 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 5220 1 (𝑎𝑇𝑏 ↔ ((𝐹𝑎)𝑅(𝐹𝑏) ∨ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wo 878   = wceq 1656  csb 3757   class class class wbr 4875  {copab 4937  cfv 6127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-iota 6090  df-fv 6135
This theorem is referenced by:  fnwe2lem2  38459  fnwe2lem3  38460
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