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Theorem bnj124 31458
Description: Technical lemma for bnj150 31463. 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 3689 . . 3 ([𝐹 / 𝑓]𝜁′[𝐹 / 𝑓]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜𝜑′𝜓′)))
4 bnj124.1 . . . . 5 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
54bnj95 31451 . . . 4 𝐹 ∈ V
6 nfv 2010 . . . . 5 𝑓(𝑅 FrSe 𝐴𝑥𝐴)
76sbc19.21g 3698 . . . 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 6190 . . . . . . . 8 (𝑓 = 𝑧 → (𝑓 Fn 1𝑜𝑧 Fn 1𝑜))
10 fneq1 6190 . . . . . . . 8 (𝑧 = 𝐹 → (𝑧 Fn 1𝑜𝐹 Fn 1𝑜))
119, 10sbcie2g 3667 . . . . . . 7 (𝐹 ∈ V → ([𝐹 / 𝑓]𝑓 Fn 1𝑜𝐹 Fn 1𝑜))
125, 11ax-mp 5 . . . . . 6 ([𝐹 / 𝑓]𝑓 Fn 1𝑜𝐹 Fn 1𝑜)
1312bicomi 216 . . . . 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 31317 . . . 4 ([𝐹 / 𝑓](𝑓 Fn 1𝑜𝜑′𝜓′) ↔ (𝐹 Fn 1𝑜𝜑″𝜓″))
1716imbi2i 328 . . 3 (((𝑅 FrSe 𝐴𝑥𝐴) → [𝐹 / 𝑓](𝑓 Fn 1𝑜𝜑′𝜓′)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1𝑜𝜑″𝜓″)))
183, 8, 173bitri 289 . 2 ([𝐹 / 𝑓]𝜁′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1𝑜𝜑″𝜓″)))
191, 18bitri 267 1 (𝜁″ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1𝑜𝜑″𝜓″)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  Vcvv 3385  [wsbc 3633  c0 4115  {csn 4368  cop 4374   Fn wfn 6096  1𝑜c1o 7792   predc-bnj14 31274   FrSe w-bnj15 31278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-fun 6103  df-fn 6104
This theorem is referenced by:  bnj150  31463  bnj153  31467
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