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Theorem bnj976 34539
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj976.1 (𝜒 ↔ (𝑁𝐷𝑓 Fn 𝑁𝜑𝜓))
bnj976.2 (𝜑′[𝐺 / 𝑓]𝜑)
bnj976.3 (𝜓′[𝐺 / 𝑓]𝜓)
bnj976.4 (𝜒′[𝐺 / 𝑓]𝜒)
bnj976.5 𝐺 ∈ V
Assertion
Ref Expression
bnj976 (𝜒′ ↔ (𝑁𝐷𝐺 Fn 𝑁𝜑′𝜓′))
Distinct variable groups:   𝐷,𝑓   𝑓,𝑁
Allowed substitution hints:   𝜑(𝑓)   𝜓(𝑓)   𝜒(𝑓)   𝐺(𝑓)   𝜑′(𝑓)   𝜓′(𝑓)   𝜒′(𝑓)

Proof of Theorem bnj976
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 bnj976.4 . 2 (𝜒′[𝐺 / 𝑓]𝜒)
2 sbccow 3796 . 2 ([𝐺 / ][ / 𝑓]𝜒[𝐺 / 𝑓]𝜒)
3 bnj976.5 . . 3 𝐺 ∈ V
4 bnj252 34465 . . . . . 6 ((𝑁𝐷𝑓 Fn 𝑁𝜑𝜓) ↔ (𝑁𝐷 ∧ (𝑓 Fn 𝑁𝜑𝜓)))
54sbcbii 3834 . . . . 5 ([ / 𝑓](𝑁𝐷𝑓 Fn 𝑁𝜑𝜓) ↔ [ / 𝑓](𝑁𝐷 ∧ (𝑓 Fn 𝑁𝜑𝜓)))
6 bnj976.1 . . . . . 6 (𝜒 ↔ (𝑁𝐷𝑓 Fn 𝑁𝜑𝜓))
76sbcbii 3834 . . . . 5 ([ / 𝑓]𝜒[ / 𝑓](𝑁𝐷𝑓 Fn 𝑁𝜑𝜓))
8 vex 3465 . . . . . . . 8 ∈ V
98bnj525 34500 . . . . . . 7 ([ / 𝑓]𝑁𝐷𝑁𝐷)
10 sbc3an 3843 . . . . . . . 8 ([ / 𝑓](𝑓 Fn 𝑁𝜑𝜓) ↔ ([ / 𝑓]𝑓 Fn 𝑁[ / 𝑓]𝜑[ / 𝑓]𝜓))
11 bnj62 34482 . . . . . . . . 9 ([ / 𝑓]𝑓 Fn 𝑁 Fn 𝑁)
12113anbi1i 1154 . . . . . . . 8 (([ / 𝑓]𝑓 Fn 𝑁[ / 𝑓]𝜑[ / 𝑓]𝜓) ↔ ( Fn 𝑁[ / 𝑓]𝜑[ / 𝑓]𝜓))
1310, 12bitri 274 . . . . . . 7 ([ / 𝑓](𝑓 Fn 𝑁𝜑𝜓) ↔ ( Fn 𝑁[ / 𝑓]𝜑[ / 𝑓]𝜓))
149, 13anbi12i 626 . . . . . 6 (([ / 𝑓]𝑁𝐷[ / 𝑓](𝑓 Fn 𝑁𝜑𝜓)) ↔ (𝑁𝐷 ∧ ( Fn 𝑁[ / 𝑓]𝜑[ / 𝑓]𝜓)))
15 sbcan 3826 . . . . . 6 ([ / 𝑓](𝑁𝐷 ∧ (𝑓 Fn 𝑁𝜑𝜓)) ↔ ([ / 𝑓]𝑁𝐷[ / 𝑓](𝑓 Fn 𝑁𝜑𝜓)))
16 bnj252 34465 . . . . . 6 ((𝑁𝐷 Fn 𝑁[ / 𝑓]𝜑[ / 𝑓]𝜓) ↔ (𝑁𝐷 ∧ ( Fn 𝑁[ / 𝑓]𝜑[ / 𝑓]𝜓)))
1714, 15, 163bitr4ri 303 . . . . 5 ((𝑁𝐷 Fn 𝑁[ / 𝑓]𝜑[ / 𝑓]𝜓) ↔ [ / 𝑓](𝑁𝐷 ∧ (𝑓 Fn 𝑁𝜑𝜓)))
185, 7, 173bitr4i 302 . . . 4 ([ / 𝑓]𝜒 ↔ (𝑁𝐷 Fn 𝑁[ / 𝑓]𝜑[ / 𝑓]𝜓))
19 fneq1 6646 . . . . . . 7 ( = 𝐺 → ( Fn 𝑁𝐺 Fn 𝑁))
20 sbceq1a 3784 . . . . . . . 8 ( = 𝐺 → ([ / 𝑓]𝜑[𝐺 / ][ / 𝑓]𝜑))
21 bnj976.2 . . . . . . . . 9 (𝜑′[𝐺 / 𝑓]𝜑)
22 sbccow 3796 . . . . . . . . 9 ([𝐺 / ][ / 𝑓]𝜑[𝐺 / 𝑓]𝜑)
2321, 22bitr4i 277 . . . . . . . 8 (𝜑′[𝐺 / ][ / 𝑓]𝜑)
2420, 23bitr4di 288 . . . . . . 7 ( = 𝐺 → ([ / 𝑓]𝜑𝜑′))
25 sbceq1a 3784 . . . . . . . 8 ( = 𝐺 → ([ / 𝑓]𝜓[𝐺 / ][ / 𝑓]𝜓))
26 bnj976.3 . . . . . . . . 9 (𝜓′[𝐺 / 𝑓]𝜓)
27 sbccow 3796 . . . . . . . . 9 ([𝐺 / ][ / 𝑓]𝜓[𝐺 / 𝑓]𝜓)
2826, 27bitr4i 277 . . . . . . . 8 (𝜓′[𝐺 / ][ / 𝑓]𝜓)
2925, 28bitr4di 288 . . . . . . 7 ( = 𝐺 → ([ / 𝑓]𝜓𝜓′))
3019, 24, 293anbi123d 1432 . . . . . 6 ( = 𝐺 → (( Fn 𝑁[ / 𝑓]𝜑[ / 𝑓]𝜓) ↔ (𝐺 Fn 𝑁𝜑′𝜓′)))
3130anbi2d 628 . . . . 5 ( = 𝐺 → ((𝑁𝐷 ∧ ( Fn 𝑁[ / 𝑓]𝜑[ / 𝑓]𝜓)) ↔ (𝑁𝐷 ∧ (𝐺 Fn 𝑁𝜑′𝜓′))))
32 bnj252 34465 . . . . 5 ((𝑁𝐷𝐺 Fn 𝑁𝜑′𝜓′) ↔ (𝑁𝐷 ∧ (𝐺 Fn 𝑁𝜑′𝜓′)))
3331, 16, 323bitr4g 313 . . . 4 ( = 𝐺 → ((𝑁𝐷 Fn 𝑁[ / 𝑓]𝜑[ / 𝑓]𝜓) ↔ (𝑁𝐷𝐺 Fn 𝑁𝜑′𝜓′)))
3418, 33bitrid 282 . . 3 ( = 𝐺 → ([ / 𝑓]𝜒 ↔ (𝑁𝐷𝐺 Fn 𝑁𝜑′𝜓′)))
353, 34sbcie 3817 . 2 ([𝐺 / ][ / 𝑓]𝜒 ↔ (𝑁𝐷𝐺 Fn 𝑁𝜑′𝜓′))
361, 2, 353bitr2i 298 1 (𝜒′ ↔ (𝑁𝐷𝐺 Fn 𝑁𝜑′𝜓′))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3461  [wsbc 3773   Fn wfn 6544  w-bnj17 34448
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-fun 6551  df-fn 6552  df-bnj17 34449
This theorem is referenced by:  bnj910  34710  bnj999  34720  bnj907  34729
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