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Theorem bnj124 34716
Description: Technical lemma for bnj150 34721. 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.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj124.1 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
bnj124.2 (𝜑″[𝐹 / 𝑓]𝜑′)
bnj124.3 (𝜓″[𝐹 / 𝑓]𝜓′)
bnj124.4 (𝜁″[𝐹 / 𝑓]𝜁′)
bnj124.5 (𝜁′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1o𝜑′𝜓′)))
Assertion
Ref Expression
bnj124 (𝜁″ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1o𝜑″𝜓″)))
Distinct variable groups:   𝐴,𝑓   𝑅,𝑓   𝑥,𝑓
Allowed substitution hints:   𝐴(𝑥)   𝑅(𝑥)   𝐹(𝑥,𝑓)   𝜑′(𝑥,𝑓)   𝜓′(𝑥,𝑓)   𝜁′(𝑥,𝑓)   𝜑″(𝑥,𝑓)   𝜓″(𝑥,𝑓)   𝜁″(𝑥,𝑓)

Proof of Theorem bnj124
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bnj124.4 . 2 (𝜁″[𝐹 / 𝑓]𝜁′)
2 bnj124.5 . . . 4 (𝜁′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1o𝜑′𝜓′)))
32sbcbii 3837 . . 3 ([𝐹 / 𝑓]𝜁′[𝐹 / 𝑓]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1o𝜑′𝜓′)))
4 bnj124.1 . . . . 5 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
54bnj95 34709 . . . 4 𝐹 ∈ V
6 nfv 1910 . . . . 5 𝑓(𝑅 FrSe 𝐴𝑥𝐴)
76sbc19.21g 3854 . . . 4 (𝐹 ∈ V → ([𝐹 / 𝑓]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1o𝜑′𝜓′)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → [𝐹 / 𝑓](𝑓 Fn 1o𝜑′𝜓′))))
85, 7ax-mp 5 . . 3 ([𝐹 / 𝑓]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1o𝜑′𝜓′)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → [𝐹 / 𝑓](𝑓 Fn 1o𝜑′𝜓′)))
9 fneq1 6651 . . . . . . . 8 (𝑓 = 𝑧 → (𝑓 Fn 1o𝑧 Fn 1o))
10 fneq1 6651 . . . . . . . 8 (𝑧 = 𝐹 → (𝑧 Fn 1o𝐹 Fn 1o))
119, 10sbcie2g 3819 . . . . . . 7 (𝐹 ∈ V → ([𝐹 / 𝑓]𝑓 Fn 1o𝐹 Fn 1o))
125, 11ax-mp 5 . . . . . 6 ([𝐹 / 𝑓]𝑓 Fn 1o𝐹 Fn 1o)
1312bicomi 223 . . . . 5 (𝐹 Fn 1o[𝐹 / 𝑓]𝑓 Fn 1o)
14 bnj124.2 . . . . 5 (𝜑″[𝐹 / 𝑓]𝜑′)
15 bnj124.3 . . . . 5 (𝜓″[𝐹 / 𝑓]𝜓′)
1613, 14, 15, 5bnj206 34576 . . . 4 ([𝐹 / 𝑓](𝑓 Fn 1o𝜑′𝜓′) ↔ (𝐹 Fn 1o𝜑″𝜓″))
1716imbi2i 335 . . 3 (((𝑅 FrSe 𝐴𝑥𝐴) → [𝐹 / 𝑓](𝑓 Fn 1o𝜑′𝜓′)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1o𝜑″𝜓″)))
183, 8, 173bitri 296 . 2 ([𝐹 / 𝑓]𝜁′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1o𝜑″𝜓″)))
191, 18bitri 274 1 (𝜁″ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1o𝜑″𝜓″)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  Vcvv 3462  [wsbc 3776  c0 4325  {csn 4633  cop 4639   Fn wfn 6549  1oc1o 8489   predc-bnj14 34533   FrSe w-bnj15 34537
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-10 2130  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-fun 6556  df-fn 6557
This theorem is referenced by:  bnj150  34721  bnj153  34725
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