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Theorem bnj581 32888
Description: Technical lemma for bnj580 32893. 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.) Remove unnecessary distinct variable conditions. (Revised by Andrew Salmon, 9-Jul-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj581.3 (𝜒 ↔ (𝑓 Fn 𝑛𝜑𝜓))
bnj581.4 (𝜑′[𝑔 / 𝑓]𝜑)
bnj581.5 (𝜓′[𝑔 / 𝑓]𝜓)
bnj581.6 (𝜒′[𝑔 / 𝑓]𝜒)
Assertion
Ref Expression
bnj581 (𝜒′ ↔ (𝑔 Fn 𝑛𝜑′𝜓′))
Distinct variable group:   𝑓,𝑛
Allowed substitution hints:   𝜑(𝑓,𝑔,𝑛)   𝜓(𝑓,𝑔,𝑛)   𝜒(𝑓,𝑔,𝑛)   𝜑′(𝑓,𝑔,𝑛)   𝜓′(𝑓,𝑔,𝑛)   𝜒′(𝑓,𝑔,𝑛)

Proof of Theorem bnj581
StepHypRef Expression
1 bnj581.6 . 2 (𝜒′[𝑔 / 𝑓]𝜒)
2 bnj581.3 . . 3 (𝜒 ↔ (𝑓 Fn 𝑛𝜑𝜓))
32sbcbii 3776 . 2 ([𝑔 / 𝑓]𝜒[𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓))
4 sbc3an 3786 . . 3 ([𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓) ↔ ([𝑔 / 𝑓]𝑓 Fn 𝑛[𝑔 / 𝑓]𝜑[𝑔 / 𝑓]𝜓))
5 bnj62 32699 . . . . 5 ([𝑔 / 𝑓]𝑓 Fn 𝑛𝑔 Fn 𝑛)
65bicomi 223 . . . 4 (𝑔 Fn 𝑛[𝑔 / 𝑓]𝑓 Fn 𝑛)
7 bnj581.4 . . . 4 (𝜑′[𝑔 / 𝑓]𝜑)
8 bnj581.5 . . . 4 (𝜓′[𝑔 / 𝑓]𝜓)
96, 7, 83anbi123i 1154 . . 3 ((𝑔 Fn 𝑛𝜑′𝜓′) ↔ ([𝑔 / 𝑓]𝑓 Fn 𝑛[𝑔 / 𝑓]𝜑[𝑔 / 𝑓]𝜓))
104, 9bitr4i 277 . 2 ([𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑔 Fn 𝑛𝜑′𝜓′))
111, 3, 103bitri 297 1 (𝜒′ ↔ (𝑔 Fn 𝑛𝜑′𝜓′))
Colors of variables: wff setvar class
Syntax hints:  wb 205  w3a 1086  [wsbc 3716   Fn wfn 6428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-fun 6435  df-fn 6436
This theorem is referenced by:  bnj580  32893  bnj849  32905
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