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Theorem bnj976 29905
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 sbcco 3421 . 2 ([𝐺 / ][ / 𝑓]𝜒[𝐺 / 𝑓]𝜒)
3 bnj976.5 . . 3 𝐺 ∈ V
4 bnj252 29825 . . . . . 6 ((𝑁𝐷𝑓 Fn 𝑁𝜑𝜓) ↔ (𝑁𝐷 ∧ (𝑓 Fn 𝑁𝜑𝜓)))
54sbcbii 3454 . . . . 5 ([ / 𝑓](𝑁𝐷𝑓 Fn 𝑁𝜑𝜓) ↔ [ / 𝑓](𝑁𝐷 ∧ (𝑓 Fn 𝑁𝜑𝜓)))
6 bnj976.1 . . . . . 6 (𝜒 ↔ (𝑁𝐷𝑓 Fn 𝑁𝜑𝜓))
76sbcbii 3454 . . . . 5 ([ / 𝑓]𝜒[ / 𝑓](𝑁𝐷𝑓 Fn 𝑁𝜑𝜓))
8 vex 3172 . . . . . . . 8 ∈ V
98bnj525 29864 . . . . . . 7 ([ / 𝑓]𝑁𝐷𝑁𝐷)
10 sbc3an 3457 . . . . . . . 8 ([ / 𝑓](𝑓 Fn 𝑁𝜑𝜓) ↔ ([ / 𝑓]𝑓 Fn 𝑁[ / 𝑓]𝜑[ / 𝑓]𝜓))
11 bnj62 29843 . . . . . . . . 9 ([ / 𝑓]𝑓 Fn 𝑁 Fn 𝑁)
12113anbi1i 1245 . . . . . . . 8 (([ / 𝑓]𝑓 Fn 𝑁[ / 𝑓]𝜑[ / 𝑓]𝜓) ↔ ( Fn 𝑁[ / 𝑓]𝜑[ / 𝑓]𝜓))
1310, 12bitri 262 . . . . . . 7 ([ / 𝑓](𝑓 Fn 𝑁𝜑𝜓) ↔ ( Fn 𝑁[ / 𝑓]𝜑[ / 𝑓]𝜓))
149, 13anbi12i 728 . . . . . 6 (([ / 𝑓]𝑁𝐷[ / 𝑓](𝑓 Fn 𝑁𝜑𝜓)) ↔ (𝑁𝐷 ∧ ( Fn 𝑁[ / 𝑓]𝜑[ / 𝑓]𝜓)))
15 sbcan 3441 . . . . . 6 ([ / 𝑓](𝑁𝐷 ∧ (𝑓 Fn 𝑁𝜑𝜓)) ↔ ([ / 𝑓]𝑁𝐷[ / 𝑓](𝑓 Fn 𝑁𝜑𝜓)))
16 bnj252 29825 . . . . . 6 ((𝑁𝐷 Fn 𝑁[ / 𝑓]𝜑[ / 𝑓]𝜓) ↔ (𝑁𝐷 ∧ ( Fn 𝑁[ / 𝑓]𝜑[ / 𝑓]𝜓)))
1714, 15, 163bitr4ri 291 . . . . 5 ((𝑁𝐷 Fn 𝑁[ / 𝑓]𝜑[ / 𝑓]𝜓) ↔ [ / 𝑓](𝑁𝐷 ∧ (𝑓 Fn 𝑁𝜑𝜓)))
185, 7, 173bitr4i 290 . . . 4 ([ / 𝑓]𝜒 ↔ (𝑁𝐷 Fn 𝑁[ / 𝑓]𝜑[ / 𝑓]𝜓))
19 fneq1 5876 . . . . . . 7 ( = 𝐺 → ( Fn 𝑁𝐺 Fn 𝑁))
20 sbceq1a 3409 . . . . . . . 8 ( = 𝐺 → ([ / 𝑓]𝜑[𝐺 / ][ / 𝑓]𝜑))
21 bnj976.2 . . . . . . . . 9 (𝜑′[𝐺 / 𝑓]𝜑)
22 sbcco 3421 . . . . . . . . 9 ([𝐺 / ][ / 𝑓]𝜑[𝐺 / 𝑓]𝜑)
2321, 22bitr4i 265 . . . . . . . 8 (𝜑′[𝐺 / ][ / 𝑓]𝜑)
2420, 23syl6bbr 276 . . . . . . 7 ( = 𝐺 → ([ / 𝑓]𝜑𝜑′))
25 sbceq1a 3409 . . . . . . . 8 ( = 𝐺 → ([ / 𝑓]𝜓[𝐺 / ][ / 𝑓]𝜓))
26 bnj976.3 . . . . . . . . 9 (𝜓′[𝐺 / 𝑓]𝜓)
27 sbcco 3421 . . . . . . . . 9 ([𝐺 / ][ / 𝑓]𝜓[𝐺 / 𝑓]𝜓)
2826, 27bitr4i 265 . . . . . . . 8 (𝜓′[𝐺 / ][ / 𝑓]𝜓)
2925, 28syl6bbr 276 . . . . . . 7 ( = 𝐺 → ([ / 𝑓]𝜓𝜓′))
3019, 24, 293anbi123d 1390 . . . . . 6 ( = 𝐺 → (( Fn 𝑁[ / 𝑓]𝜑[ / 𝑓]𝜓) ↔ (𝐺 Fn 𝑁𝜑′𝜓′)))
3130anbi2d 735 . . . . 5 ( = 𝐺 → ((𝑁𝐷 ∧ ( Fn 𝑁[ / 𝑓]𝜑[ / 𝑓]𝜓)) ↔ (𝑁𝐷 ∧ (𝐺 Fn 𝑁𝜑′𝜓′))))
32 bnj252 29825 . . . . 5 ((𝑁𝐷𝐺 Fn 𝑁𝜑′𝜓′) ↔ (𝑁𝐷 ∧ (𝐺 Fn 𝑁𝜑′𝜓′)))
3331, 16, 323bitr4g 301 . . . 4 ( = 𝐺 → ((𝑁𝐷 Fn 𝑁[ / 𝑓]𝜑[ / 𝑓]𝜓) ↔ (𝑁𝐷𝐺 Fn 𝑁𝜑′𝜓′)))
3418, 33syl5bb 270 . . 3 ( = 𝐺 → ([ / 𝑓]𝜒 ↔ (𝑁𝐷𝐺 Fn 𝑁𝜑′𝜓′)))
353, 34sbcie 3433 . 2 ([𝐺 / ][ / 𝑓]𝜒 ↔ (𝑁𝐷𝐺 Fn 𝑁𝜑′𝜓′))
361, 2, 353bitr2i 286 1 (𝜒′ ↔ (𝑁𝐷𝐺 Fn 𝑁𝜑′𝜓′))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  Vcvv 3169  [wsbc 3398   Fn wfn 5782  w-bnj17 29808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-rab 2901  df-v 3171  df-sbc 3399  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-br 4575  df-opab 4635  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-fun 5789  df-fn 5790  df-bnj17 29809
This theorem is referenced by:  bnj910  30075  bnj999  30084  bnj907  30092
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