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Mirrors > Home > ILE Home > Th. List > ceqex | Unicode version |
Description: Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.) |
Ref | Expression |
---|---|
ceqex |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.8a 1601 |
. . 3
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2 | isset 2758 |
. . 3
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3 | 1, 2 | sylibr 134 |
. 2
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4 | eqeq2 2199 |
. . . 4
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5 | 4 | anbi1d 465 |
. . . . . 6
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6 | 5 | exbidv 1836 |
. . . . 5
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7 | 6 | bibi2d 232 |
. . . 4
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8 | 4, 7 | imbi12d 234 |
. . 3
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9 | 19.8a 1601 |
. . . . 5
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10 | 9 | ex 115 |
. . . 4
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11 | vex 2755 |
. . . . . 6
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12 | 11 | alexeq 2878 |
. . . . 5
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13 | sp 1522 |
. . . . . 6
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14 | 13 | com12 30 |
. . . . 5
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15 | 12, 14 | biimtrrid 153 |
. . . 4
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16 | 10, 15 | impbid 129 |
. . 3
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17 | 8, 16 | vtoclg 2812 |
. 2
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18 | 3, 17 | mpcom 36 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 |
This theorem is referenced by: ceqsexg 2880 sbc6g 3002 |
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