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Theorem bnj581 31206
Description: Technical lemma for bnj580 31211. 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). (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 9-Jul-2011.) (Contributed by Jonathan Ben-Naim, 3-Jun-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 3597 . 2 ([𝑔 / 𝑓]𝜒[𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓))
4 sbc3an 3600 . . 3 ([𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓) ↔ ([𝑔 / 𝑓]𝑓 Fn 𝑛[𝑔 / 𝑓]𝜑[𝑔 / 𝑓]𝜓))
5 bnj62 31016 . . . . 5 ([𝑔 / 𝑓]𝑓 Fn 𝑛𝑔 Fn 𝑛)
65bicomi 214 . . . 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 1388 . . 3 ((𝑔 Fn 𝑛𝜑′𝜓′) ↔ ([𝑔 / 𝑓]𝑓 Fn 𝑛[𝑔 / 𝑓]𝜑[𝑔 / 𝑓]𝜓))
104, 9bitr4i 267 . 2 ([𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑔 Fn 𝑛𝜑′𝜓′))
111, 3, 103bitri 286 1 (𝜒′ ↔ (𝑔 Fn 𝑛𝜑′𝜓′))
Colors of variables: wff setvar class
Syntax hints:  wb 196  w3a 1072  [wsbc 3541   Fn wfn 5996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-fun 6003  df-fn 6004
This theorem is referenced by:  bnj580  31211  bnj849  31223
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