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Theorem bnj1040 31378
 Description: Technical lemma for bnj69 31416. 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 3643 . 2 ([𝑗 / 𝑖]𝜒[𝑗 / 𝑖](𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
4 df-bnj17 31093 . . 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 3354 . . . . . 6 𝑗 ∈ V
65bnj525 31145 . . . . 5 ([𝑗 / 𝑖]𝑛𝐷𝑛𝐷)
76bicomi 214 . . . 4 (𝑛𝐷[𝑗 / 𝑖]𝑛𝐷)
85bnj525 31145 . . . . 5 ([𝑗 / 𝑖]𝑓 Fn 𝑛𝑓 Fn 𝑛)
98bicomi 214 . . . 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 31173 . . 3 ((𝑛𝐷𝑓 Fn 𝑛𝜑′𝜓′) ↔ ([𝑗 / 𝑖]𝑛𝐷[𝑗 / 𝑖]𝑓 Fn 𝑛[𝑗 / 𝑖]𝜑[𝑗 / 𝑖]𝜓))
13 df-bnj17 31093 . . . . 5 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) ↔ ((𝑛𝐷𝑓 Fn 𝑛𝜑) ∧ 𝜓))
1413sbcbii 3643 . . . 4 ([𝑗 / 𝑖](𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) ↔ [𝑗 / 𝑖]((𝑛𝐷𝑓 Fn 𝑛𝜑) ∧ 𝜓))
15 sbcan 3630 . . . 4 ([𝑗 / 𝑖]((𝑛𝐷𝑓 Fn 𝑛𝜑) ∧ 𝜓) ↔ ([𝑗 / 𝑖](𝑛𝐷𝑓 Fn 𝑛𝜑) ∧ [𝑗 / 𝑖]𝜓))
16 sbc3an 3645 . . . . 5 ([𝑗 / 𝑖](𝑛𝐷𝑓 Fn 𝑛𝜑) ↔ ([𝑗 / 𝑖]𝑛𝐷[𝑗 / 𝑖]𝑓 Fn 𝑛[𝑗 / 𝑖]𝜑))
1716anbi1i 610 . . . 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 286 . . 3 ([𝑗 / 𝑖](𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) ↔ (([𝑗 / 𝑖]𝑛𝐷[𝑗 / 𝑖]𝑓 Fn 𝑛[𝑗 / 𝑖]𝜑) ∧ [𝑗 / 𝑖]𝜓))
194, 12, 183bitr4ri 293 . 2 ([𝑗 / 𝑖](𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑′𝜓′))
201, 3, 193bitri 286 1 (𝜒′ ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑′𝜓′))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   ∈ wcel 2145  [wsbc 3587   Fn wfn 6026   ∧ w-bnj17 31092 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-v 3353  df-sbc 3588  df-bnj17 31093 This theorem is referenced by:  bnj1128  31396
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