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Theorem bnj919 31166
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 3633 . 2 ([𝑃 / 𝑛]𝜒[𝑃 / 𝑛](𝑛𝐷𝐹 Fn 𝑛𝜑𝜓))
4 bnj919.5 . . 3 𝑃 ∈ V
5 df-bnj17 31084 . . . . 5 ((𝑃𝐷𝐹 Fn 𝑃𝜑′𝜓′) ↔ ((𝑃𝐷𝐹 Fn 𝑃𝜑′) ∧ 𝜓′))
6 nfv 1993 . . . . . . 7 𝑛 𝑃𝐷
7 nfv 1993 . . . . . . 7 𝑛 𝐹 Fn 𝑃
8 bnj919.2 . . . . . . . 8 (𝜑′[𝑃 / 𝑛]𝜑)
9 nfsbc1v 3597 . . . . . . . 8 𝑛[𝑃 / 𝑛]𝜑
108, 9nfxfr 1928 . . . . . . 7 𝑛𝜑′
116, 7, 10nf3an 1981 . . . . . 6 𝑛(𝑃𝐷𝐹 Fn 𝑃𝜑′)
12 bnj919.3 . . . . . . 7 (𝜓′[𝑃 / 𝑛]𝜓)
13 nfsbc1v 3597 . . . . . . 7 𝑛[𝑃 / 𝑛]𝜓
1412, 13nfxfr 1928 . . . . . 6 𝑛𝜓′
1511, 14nfan 1978 . . . . 5 𝑛((𝑃𝐷𝐹 Fn 𝑃𝜑′) ∧ 𝜓′)
165, 15nfxfr 1928 . . . 4 𝑛(𝑃𝐷𝐹 Fn 𝑃𝜑′𝜓′)
17 eleq1 2828 . . . . . 6 (𝑛 = 𝑃 → (𝑛𝐷𝑃𝐷))
18 fneq2 6142 . . . . . . 7 (𝑛 = 𝑃 → (𝐹 Fn 𝑛𝐹 Fn 𝑃))
19 sbceq1a 3588 . . . . . . . 8 (𝑛 = 𝑃 → (𝜑[𝑃 / 𝑛]𝜑))
2019, 8syl6bbr 278 . . . . . . 7 (𝑛 = 𝑃 → (𝜑𝜑′))
21 sbceq1a 3588 . . . . . . . 8 (𝑛 = 𝑃 → (𝜓[𝑃 / 𝑛]𝜓))
2221, 12syl6bbr 278 . . . . . . 7 (𝑛 = 𝑃 → (𝜓𝜓′))
2318, 20, 223anbi123d 1548 . . . . . 6 (𝑛 = 𝑃 → ((𝐹 Fn 𝑛𝜑𝜓) ↔ (𝐹 Fn 𝑃𝜑′𝜓′)))
2417, 23anbi12d 749 . . . . 5 (𝑛 = 𝑃 → ((𝑛𝐷 ∧ (𝐹 Fn 𝑛𝜑𝜓)) ↔ (𝑃𝐷 ∧ (𝐹 Fn 𝑃𝜑′𝜓′))))
25 bnj252 31100 . . . . 5 ((𝑛𝐷𝐹 Fn 𝑛𝜑𝜓) ↔ (𝑛𝐷 ∧ (𝐹 Fn 𝑛𝜑𝜓)))
26 bnj252 31100 . . . . 5 ((𝑃𝐷𝐹 Fn 𝑃𝜑′𝜓′) ↔ (𝑃𝐷 ∧ (𝐹 Fn 𝑃𝜑′𝜓′)))
2724, 25, 263bitr4g 303 . . . 4 (𝑛 = 𝑃 → ((𝑛𝐷𝐹 Fn 𝑛𝜑𝜓) ↔ (𝑃𝐷𝐹 Fn 𝑃𝜑′𝜓′)))
2816, 27sbciegf 3609 . . 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 286 1 (𝜒′ ↔ (𝑃𝐷𝐹 Fn 𝑃𝜑′𝜓′))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2140  Vcvv 3341  [wsbc 3577   Fn wfn 6045  w-bnj17 31083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-v 3343  df-sbc 3578  df-fn 6053  df-bnj17 31084
This theorem is referenced by:  bnj910  31347  bnj999  31356  bnj907  31364
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