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Mirrors > Home > ILE Home > Th. List > sbcfng | Unicode version |
Description: Distribute proper substitution through the function predicate with a domain. (Contributed by Alexander van der Vekens, 15-Jul-2018.) |
Ref | Expression |
---|---|
sbcfng |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fn 5217 |
. . . 4
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2 | 1 | a1i 9 |
. . 3
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3 | 2 | sbcbidv 3021 |
. 2
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4 | sbcfung 5238 |
. . . 4
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5 | sbceqg 3073 |
. . . . 5
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6 | csbdmg 4819 |
. . . . . 6
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7 | 6 | eqeq1d 2186 |
. . . . 5
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8 | 5, 7 | bitrd 188 |
. . . 4
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9 | 4, 8 | anbi12d 473 |
. . 3
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10 | sbcan 3005 |
. . 3
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11 | df-fn 5217 |
. . 3
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12 | 9, 10, 11 | 3bitr4g 223 |
. 2
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13 | 3, 12 | bitrd 188 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4003 df-opab 4064 df-id 4292 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-fun 5216 df-fn 5217 |
This theorem is referenced by: sbcfg 5362 |
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