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Theorem bnj124 31279
 Description: Technical lemma for bnj150 31284. 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 3643 . . 3 ([𝐹 / 𝑓]𝜁′[𝐹 / 𝑓]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜𝜑′𝜓′)))
4 bnj124.1 . . . . 5 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
54bnj95 31272 . . . 4 𝐹 ∈ V
6 nfv 1995 . . . . 5 𝑓(𝑅 FrSe 𝐴𝑥𝐴)
76sbc19.21g 3652 . . . 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 6119 . . . . . . . 8 (𝑓 = 𝑧 → (𝑓 Fn 1𝑜𝑧 Fn 1𝑜))
10 fneq1 6119 . . . . . . . 8 (𝑧 = 𝐹 → (𝑧 Fn 1𝑜𝐹 Fn 1𝑜))
119, 10sbcie2g 3621 . . . . . . 7 (𝐹 ∈ V → ([𝐹 / 𝑓]𝑓 Fn 1𝑜𝐹 Fn 1𝑜))
125, 11ax-mp 5 . . . . . 6 ([𝐹 / 𝑓]𝑓 Fn 1𝑜𝐹 Fn 1𝑜)
1312bicomi 214 . . . . 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 31137 . . . 4 ([𝐹 / 𝑓](𝑓 Fn 1𝑜𝜑′𝜓′) ↔ (𝐹 Fn 1𝑜𝜑″𝜓″))
1716imbi2i 325 . . 3 (((𝑅 FrSe 𝐴𝑥𝐴) → [𝐹 / 𝑓](𝑓 Fn 1𝑜𝜑′𝜓′)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1𝑜𝜑″𝜓″)))
183, 8, 173bitri 286 . 2 ([𝐹 / 𝑓]𝜁′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1𝑜𝜑″𝜓″)))
191, 18bitri 264 1 (𝜁″ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1𝑜𝜑″𝜓″)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  Vcvv 3351  [wsbc 3587  ∅c0 4063  {csn 4316  ⟨cop 4322   Fn wfn 6026  1𝑜c1o 7706   predc-bnj14 31094   FrSe w-bnj15 31098 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-fun 6033  df-fn 6034 This theorem is referenced by:  bnj150  31284  bnj153  31288
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