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Theorem bnj873 30248
 Description: Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj873.4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj873.7 (𝜑′[𝑔 / 𝑓]𝜑)
bnj873.8 (𝜓′[𝑔 / 𝑓]𝜓)
Assertion
Ref Expression
bnj873 𝐵 = {𝑔 ∣ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)}
Distinct variable groups:   𝐷,𝑓,𝑔   𝑓,𝑛,𝑔   𝜑,𝑔   𝜓,𝑔
Allowed substitution hints:   𝜑(𝑓,𝑛)   𝜓(𝑓,𝑛)   𝐵(𝑓,𝑔,𝑛)   𝐷(𝑛)   𝜑′(𝑓,𝑔,𝑛)   𝜓′(𝑓,𝑔,𝑛)

Proof of Theorem bnj873
StepHypRef Expression
1 bnj873.4 . 2 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
2 nfv 1830 . . 3 𝑔𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)
3 nfcv 2751 . . . 4 𝑓𝐷
4 nfv 1830 . . . . 5 𝑓 𝑔 Fn 𝑛
5 bnj873.7 . . . . . 6 (𝜑′[𝑔 / 𝑓]𝜑)
6 nfsbc1v 3422 . . . . . 6 𝑓[𝑔 / 𝑓]𝜑
75, 6nfxfr 1771 . . . . 5 𝑓𝜑′
8 bnj873.8 . . . . . 6 (𝜓′[𝑔 / 𝑓]𝜓)
9 nfsbc1v 3422 . . . . . 6 𝑓[𝑔 / 𝑓]𝜓
108, 9nfxfr 1771 . . . . 5 𝑓𝜓′
114, 7, 10nf3an 1819 . . . 4 𝑓(𝑔 Fn 𝑛𝜑′𝜓′)
123, 11nfrex 2990 . . 3 𝑓𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)
13 fneq1 5893 . . . . 5 (𝑓 = 𝑔 → (𝑓 Fn 𝑛𝑔 Fn 𝑛))
14 sbceq1a 3413 . . . . . 6 (𝑓 = 𝑔 → (𝜑[𝑔 / 𝑓]𝜑))
1514, 5syl6bbr 277 . . . . 5 (𝑓 = 𝑔 → (𝜑𝜑′))
16 sbceq1a 3413 . . . . . 6 (𝑓 = 𝑔 → (𝜓[𝑔 / 𝑓]𝜓))
1716, 8syl6bbr 277 . . . . 5 (𝑓 = 𝑔 → (𝜓𝜓′))
1813, 15, 173anbi123d 1391 . . . 4 (𝑓 = 𝑔 → ((𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑔 Fn 𝑛𝜑′𝜓′)))
1918rexbidv 3034 . . 3 (𝑓 = 𝑔 → (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)))
202, 12, 19cbvab 2733 . 2 {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)} = {𝑔 ∣ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)}
211, 20eqtri 2632 1 𝐵 = {𝑔 ∣ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)}
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ w3a 1031   = wceq 1475  {cab 2596  ∃wrex 2897  [wsbc 3402   Fn wfn 5799 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-fun 5806  df-fn 5807 This theorem is referenced by:  bnj849  30249  bnj893  30252
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