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Theorem bnj581 32066
Description: Technical lemma for bnj580 32071. 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 3833 . 2 ([𝑔 / 𝑓]𝜒[𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓))
4 sbc3an 3842 . . 3 ([𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓) ↔ ([𝑔 / 𝑓]𝑓 Fn 𝑛[𝑔 / 𝑓]𝜑[𝑔 / 𝑓]𝜓))
5 bnj62 31876 . . . . 5 ([𝑔 / 𝑓]𝑓 Fn 𝑛𝑔 Fn 𝑛)
65bicomi 225 . . . 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 1149 . . 3 ((𝑔 Fn 𝑛𝜑′𝜓′) ↔ ([𝑔 / 𝑓]𝑓 Fn 𝑛[𝑔 / 𝑓]𝜑[𝑔 / 𝑓]𝜓))
104, 9bitr4i 279 . 2 ([𝑔 / 𝑓](𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑔 Fn 𝑛𝜑′𝜓′))
111, 3, 103bitri 298 1 (𝜒′ ↔ (𝑔 Fn 𝑛𝜑′𝜓′))
Colors of variables: wff setvar class
Syntax hints:  wb 207  w3a 1081  [wsbc 3776   Fn wfn 6347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-fun 6354  df-fn 6355
This theorem is referenced by:  bnj580  32071  bnj849  32083
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