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Theorem bnj1040 31363
Description: Technical lemma for bnj69 31401. 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.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1040.1 (𝜑′[𝑗 / 𝑖]𝜑)
bnj1040.2 (𝜓′[𝑗 / 𝑖]𝜓)
bnj1040.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1040.4 (𝜒′[𝑗 / 𝑖]𝜒)
Assertion
Ref Expression
bnj1040 (𝜒′ ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑′𝜓′))
Distinct variable groups:   𝐷,𝑖   𝑓,𝑖   𝑖,𝑛
Allowed substitution hints:   𝜑(𝑓,𝑖,𝑗,𝑛)   𝜓(𝑓,𝑖,𝑗,𝑛)   𝜒(𝑓,𝑖,𝑗,𝑛)   𝐷(𝑓,𝑗,𝑛)   𝜑′(𝑓,𝑖,𝑗,𝑛)   𝜓′(𝑓,𝑖,𝑗,𝑛)   𝜒′(𝑓,𝑖,𝑗,𝑛)

Proof of Theorem bnj1040
StepHypRef Expression
1 bnj1040.4 . 2 (𝜒′[𝑗 / 𝑖]𝜒)
2 bnj1040.3 . . 3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
32sbcbii 3689 . 2 ([𝑗 / 𝑖]𝜒[𝑗 / 𝑖](𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
4 df-bnj17 31078 . . 3 (([𝑗 / 𝑖]𝑛𝐷[𝑗 / 𝑖]𝑓 Fn 𝑛[𝑗 / 𝑖]𝜑[𝑗 / 𝑖]𝜓) ↔ (([𝑗 / 𝑖]𝑛𝐷[𝑗 / 𝑖]𝑓 Fn 𝑛[𝑗 / 𝑖]𝜑) ∧ [𝑗 / 𝑖]𝜓))
5 vex 3394 . . . . . 6 𝑗 ∈ V
65bnj525 31130 . . . . 5 ([𝑗 / 𝑖]𝑛𝐷𝑛𝐷)
76bicomi 215 . . . 4 (𝑛𝐷[𝑗 / 𝑖]𝑛𝐷)
85bnj525 31130 . . . . 5 ([𝑗 / 𝑖]𝑓 Fn 𝑛𝑓 Fn 𝑛)
98bicomi 215 . . . 4 (𝑓 Fn 𝑛[𝑗 / 𝑖]𝑓 Fn 𝑛)
10 bnj1040.1 . . . 4 (𝜑′[𝑗 / 𝑖]𝜑)
11 bnj1040.2 . . . 4 (𝜓′[𝑗 / 𝑖]𝜓)
127, 9, 10, 11bnj887 31158 . . 3 ((𝑛𝐷𝑓 Fn 𝑛𝜑′𝜓′) ↔ ([𝑗 / 𝑖]𝑛𝐷[𝑗 / 𝑖]𝑓 Fn 𝑛[𝑗 / 𝑖]𝜑[𝑗 / 𝑖]𝜓))
13 df-bnj17 31078 . . . . 5 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) ↔ ((𝑛𝐷𝑓 Fn 𝑛𝜑) ∧ 𝜓))
1413sbcbii 3689 . . . 4 ([𝑗 / 𝑖](𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) ↔ [𝑗 / 𝑖]((𝑛𝐷𝑓 Fn 𝑛𝜑) ∧ 𝜓))
15 sbcan 3676 . . . 4 ([𝑗 / 𝑖]((𝑛𝐷𝑓 Fn 𝑛𝜑) ∧ 𝜓) ↔ ([𝑗 / 𝑖](𝑛𝐷𝑓 Fn 𝑛𝜑) ∧ [𝑗 / 𝑖]𝜓))
16 sbc3an 3691 . . . . 5 ([𝑗 / 𝑖](𝑛𝐷𝑓 Fn 𝑛𝜑) ↔ ([𝑗 / 𝑖]𝑛𝐷[𝑗 / 𝑖]𝑓 Fn 𝑛[𝑗 / 𝑖]𝜑))
1716anbi1i 612 . . . 4 (([𝑗 / 𝑖](𝑛𝐷𝑓 Fn 𝑛𝜑) ∧ [𝑗 / 𝑖]𝜓) ↔ (([𝑗 / 𝑖]𝑛𝐷[𝑗 / 𝑖]𝑓 Fn 𝑛[𝑗 / 𝑖]𝜑) ∧ [𝑗 / 𝑖]𝜓))
1814, 15, 173bitri 288 . . 3 ([𝑗 / 𝑖](𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) ↔ (([𝑗 / 𝑖]𝑛𝐷[𝑗 / 𝑖]𝑓 Fn 𝑛[𝑗 / 𝑖]𝜑) ∧ [𝑗 / 𝑖]𝜓))
194, 12, 183bitr4ri 295 . 2 ([𝑗 / 𝑖](𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑′𝜓′))
201, 3, 193bitri 288 1 (𝜒′ ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑′𝜓′))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  w3a 1100  wcel 2156  [wsbc 3633   Fn wfn 6096  w-bnj17 31077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-v 3393  df-sbc 3634  df-bnj17 31078
This theorem is referenced by:  bnj1128  31381
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