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Theorem bnj1385 29141
 Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1385.1
bnj1385.2
bnj1385.3
bnj1385.4
bnj1385.5
bnj1385.6
bnj1385.7
Assertion
Ref Expression
bnj1385
Distinct variable groups:   ,,,   ,   ,   ,,,   ,
Allowed substitution hints:   (,,,)   (,,,)   ()   (,,)   (,,)   (,,)   (,,,)

Proof of Theorem bnj1385
StepHypRef Expression
1 nfv 1629 . . . . . . 7
2 bnj1385.4 . . . . . . . . . 10
32nfcii 2562 . . . . . . . . 9
43nfcri 2565 . . . . . . . 8
5 nfv 1629 . . . . . . . 8
64, 5nfim 1832 . . . . . . 7
7 eleq1 2495 . . . . . . . 8
8 funeq 5465 . . . . . . . 8
97, 8imbi12d 312 . . . . . . 7
101, 6, 9cbval 1982 . . . . . 6
11 df-ral 2702 . . . . . 6
12 df-ral 2702 . . . . . 6
1310, 11, 123bitr4i 269 . . . . 5
14 bnj1385.1 . . . . 5
15 bnj1385.5 . . . . 5
1613, 14, 153bitr4i 269 . . . 4
17 nfv 1629 . . . . . 6
18 nfv 1629 . . . . . . . 8
193, 18nfral 2751 . . . . . . 7
204, 19nfim 1832 . . . . . 6
21 dmeq 5062 . . . . . . . . . . . . 13
2221ineq1d 3533 . . . . . . . . . . . 12
23 bnj1385.2 . . . . . . . . . . . 12
24 bnj1385.6 . . . . . . . . . . . 12
2522, 23, 243eqtr4g 2492 . . . . . . . . . . 11
2625reseq2d 5138 . . . . . . . . . 10
27 reseq1 5132 . . . . . . . . . 10
2826, 27eqtrd 2467 . . . . . . . . 9
2925reseq2d 5138 . . . . . . . . 9
3028, 29eqeq12d 2449 . . . . . . . 8
3130ralbidv 2717 . . . . . . 7
327, 31imbi12d 312 . . . . . 6
3317, 20, 32cbval 1982 . . . . 5
34 df-ral 2702 . . . . 5
35 df-ral 2702 . . . . 5
3633, 34, 353bitr4i 269 . . . 4
3716, 36anbi12i 679 . . 3
38 bnj1385.3 . . 3
39 bnj1385.7 . . 3
4037, 38, 393bitr4i 269 . 2
4115, 24, 39bnj1383 29140 . 2
4240, 41sylbi 188 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wal 1549   wceq 1652   wcel 1725  wral 2697   cin 3311  cuni 4007   cdm 4870   cres 4872   wfun 5440 This theorem is referenced by:  bnj1386  29142 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-res 4882  df-iota 5410  df-fun 5448  df-fv 5454
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