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Theorem bnj124 32048
Description: Technical lemma for bnj150 32053. 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 3833 . . 3 ([𝐹 / 𝑓]𝜁′[𝐹 / 𝑓]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1o𝜑′𝜓′)))
4 bnj124.1 . . . . 5 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
54bnj95 32041 . . . 4 𝐹 ∈ V
6 nfv 1908 . . . . 5 𝑓(𝑅 FrSe 𝐴𝑥𝐴)
76sbc19.21g 3850 . . . 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 6443 . . . . . . . 8 (𝑓 = 𝑧 → (𝑓 Fn 1o𝑧 Fn 1o))
10 fneq1 6443 . . . . . . . 8 (𝑧 = 𝐹 → (𝑧 Fn 1o𝐹 Fn 1o))
119, 10sbcie2g 3815 . . . . . . 7 (𝐹 ∈ V → ([𝐹 / 𝑓]𝑓 Fn 1o𝐹 Fn 1o))
125, 11ax-mp 5 . . . . . 6 ([𝐹 / 𝑓]𝑓 Fn 1o𝐹 Fn 1o)
1312bicomi 225 . . . . 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 31906 . . . 4 ([𝐹 / 𝑓](𝑓 Fn 1o𝜑′𝜓′) ↔ (𝐹 Fn 1o𝜑″𝜓″))
1716imbi2i 337 . . 3 (((𝑅 FrSe 𝐴𝑥𝐴) → [𝐹 / 𝑓](𝑓 Fn 1o𝜑′𝜓′)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1o𝜑″𝜓″)))
183, 8, 173bitri 298 . 2 ([𝐹 / 𝑓]𝜁′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1o𝜑″𝜓″)))
191, 18bitri 276 1 (𝜁″ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1o𝜑″𝜓″)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  Vcvv 3500  [wsbc 3776  c0 4295  {csn 4564  cop 4570   Fn wfn 6349  1oc1o 8091   predc-bnj14 31863   FrSe w-bnj15 31867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-fun 6356  df-fn 6357
This theorem is referenced by:  bnj150  32053  bnj153  32057
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