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Theorem bnj919 31925
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj919.1 (𝜒 ↔ (𝑛𝐷𝐹 Fn 𝑛𝜑𝜓))
bnj919.2 (𝜑′[𝑃 / 𝑛]𝜑)
bnj919.3 (𝜓′[𝑃 / 𝑛]𝜓)
bnj919.4 (𝜒′[𝑃 / 𝑛]𝜒)
bnj919.5 𝑃 ∈ V
Assertion
Ref Expression
bnj919 (𝜒′ ↔ (𝑃𝐷𝐹 Fn 𝑃𝜑′𝜓′))
Distinct variable groups:   𝐷,𝑛   𝑛,𝐹   𝑃,𝑛
Allowed substitution hints:   𝜑(𝑛)   𝜓(𝑛)   𝜒(𝑛)   𝜑′(𝑛)   𝜓′(𝑛)   𝜒′(𝑛)

Proof of Theorem bnj919
StepHypRef Expression
1 bnj919.4 . 2 (𝜒′[𝑃 / 𝑛]𝜒)
2 bnj919.1 . . 3 (𝜒 ↔ (𝑛𝐷𝐹 Fn 𝑛𝜑𝜓))
32sbcbii 3832 . 2 ([𝑃 / 𝑛]𝜒[𝑃 / 𝑛](𝑛𝐷𝐹 Fn 𝑛𝜑𝜓))
4 bnj919.5 . . 3 𝑃 ∈ V
5 df-bnj17 31844 . . . . 5 ((𝑃𝐷𝐹 Fn 𝑃𝜑′𝜓′) ↔ ((𝑃𝐷𝐹 Fn 𝑃𝜑′) ∧ 𝜓′))
6 nfv 1908 . . . . . . 7 𝑛 𝑃𝐷
7 nfv 1908 . . . . . . 7 𝑛 𝐹 Fn 𝑃
8 bnj919.2 . . . . . . . 8 (𝜑′[𝑃 / 𝑛]𝜑)
9 nfsbc1v 3795 . . . . . . . 8 𝑛[𝑃 / 𝑛]𝜑
108, 9nfxfr 1846 . . . . . . 7 𝑛𝜑′
116, 7, 10nf3an 1895 . . . . . 6 𝑛(𝑃𝐷𝐹 Fn 𝑃𝜑′)
12 bnj919.3 . . . . . . 7 (𝜓′[𝑃 / 𝑛]𝜓)
13 nfsbc1v 3795 . . . . . . 7 𝑛[𝑃 / 𝑛]𝜓
1412, 13nfxfr 1846 . . . . . 6 𝑛𝜓′
1511, 14nfan 1893 . . . . 5 𝑛((𝑃𝐷𝐹 Fn 𝑃𝜑′) ∧ 𝜓′)
165, 15nfxfr 1846 . . . 4 𝑛(𝑃𝐷𝐹 Fn 𝑃𝜑′𝜓′)
17 eleq1 2904 . . . . . 6 (𝑛 = 𝑃 → (𝑛𝐷𝑃𝐷))
18 fneq2 6441 . . . . . . 7 (𝑛 = 𝑃 → (𝐹 Fn 𝑛𝐹 Fn 𝑃))
19 sbceq1a 3786 . . . . . . . 8 (𝑛 = 𝑃 → (𝜑[𝑃 / 𝑛]𝜑))
2019, 8syl6bbr 290 . . . . . . 7 (𝑛 = 𝑃 → (𝜑𝜑′))
21 sbceq1a 3786 . . . . . . . 8 (𝑛 = 𝑃 → (𝜓[𝑃 / 𝑛]𝜓))
2221, 12syl6bbr 290 . . . . . . 7 (𝑛 = 𝑃 → (𝜓𝜓′))
2318, 20, 223anbi123d 1429 . . . . . 6 (𝑛 = 𝑃 → ((𝐹 Fn 𝑛𝜑𝜓) ↔ (𝐹 Fn 𝑃𝜑′𝜓′)))
2417, 23anbi12d 630 . . . . 5 (𝑛 = 𝑃 → ((𝑛𝐷 ∧ (𝐹 Fn 𝑛𝜑𝜓)) ↔ (𝑃𝐷 ∧ (𝐹 Fn 𝑃𝜑′𝜓′))))
25 bnj252 31860 . . . . 5 ((𝑛𝐷𝐹 Fn 𝑛𝜑𝜓) ↔ (𝑛𝐷 ∧ (𝐹 Fn 𝑛𝜑𝜓)))
26 bnj252 31860 . . . . 5 ((𝑃𝐷𝐹 Fn 𝑃𝜑′𝜓′) ↔ (𝑃𝐷 ∧ (𝐹 Fn 𝑃𝜑′𝜓′)))
2724, 25, 263bitr4g 315 . . . 4 (𝑛 = 𝑃 → ((𝑛𝐷𝐹 Fn 𝑛𝜑𝜓) ↔ (𝑃𝐷𝐹 Fn 𝑃𝜑′𝜓′)))
2816, 27sbciegf 3812 . . 3 (𝑃 ∈ V → ([𝑃 / 𝑛](𝑛𝐷𝐹 Fn 𝑛𝜑𝜓) ↔ (𝑃𝐷𝐹 Fn 𝑃𝜑′𝜓′)))
294, 28ax-mp 5 . 2 ([𝑃 / 𝑛](𝑛𝐷𝐹 Fn 𝑛𝜑𝜓) ↔ (𝑃𝐷𝐹 Fn 𝑃𝜑′𝜓′))
301, 3, 293bitri 298 1 (𝜒′ ↔ (𝑃𝐷𝐹 Fn 𝑃𝜑′𝜓′))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2106  Vcvv 3499  [wsbc 3775   Fn wfn 6346  w-bnj17 31843
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 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796
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 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-v 3501  df-sbc 3776  df-fn 6354  df-bnj17 31844
This theorem is referenced by:  bnj910  32107  bnj999  32116  bnj907  32124
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