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Theorem bnj581 31495
Description: Technical lemma for bnj580 31500. 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 3689 . 2 ([𝑔 / 𝑓]𝜒[𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓))
4 sbc3an 3691 . . 3 ([𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓) ↔ ([𝑔 / 𝑓]𝑓 Fn 𝑛[𝑔 / 𝑓]𝜑[𝑔 / 𝑓]𝜓))
5 bnj62 31306 . . . . 5 ([𝑔 / 𝑓]𝑓 Fn 𝑛𝑔 Fn 𝑛)
65bicomi 216 . . . 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 1195 . . 3 ((𝑔 Fn 𝑛𝜑′𝜓′) ↔ ([𝑔 / 𝑓]𝑓 Fn 𝑛[𝑔 / 𝑓]𝜑[𝑔 / 𝑓]𝜓))
104, 9bitr4i 270 . 2 ([𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑔 Fn 𝑛𝜑′𝜓′))
111, 3, 103bitri 289 1 (𝜒′ ↔ (𝑔 Fn 𝑛𝜑′𝜓′))
Colors of variables: wff setvar class
Syntax hints:  wb 198  w3a 1108  [wsbc 3633   Fn wfn 6096
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
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:  bnj580  31500  bnj849  31512
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