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Mirrors > Home > MPE Home > Th. List > sbccom | Structured version Visualization version Unicode version |
Description: Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.) |
Ref | Expression |
---|---|
sbccom |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbccomlem 3337 |
. . . 4
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2 | sbccomlem 3337 |
. . . . . . 7
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3 | 2 | sbcbii 3322 |
. . . . . 6
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4 | sbccomlem 3337 |
. . . . . 6
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5 | 3, 4 | bitri 253 |
. . . . 5
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6 | 5 | sbcbii 3322 |
. . . 4
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7 | sbccomlem 3337 |
. . . . 5
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8 | 7 | sbcbii 3322 |
. . . 4
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9 | 1, 6, 8 | 3bitr3i 279 |
. . 3
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10 | sbcco 3289 |
. . 3
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11 | sbcco 3289 |
. . 3
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12 | 9, 10, 11 | 3bitr3i 279 |
. 2
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13 | sbcco 3289 |
. . 3
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14 | 13 | sbcbii 3322 |
. 2
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15 | sbcco 3289 |
. . 3
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16 | 15 | sbcbii 3322 |
. 2
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17 | 12, 14, 16 | 3bitr3i 279 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-clab 2437 df-cleq 2443 df-clel 2446 df-v 3046 df-sbc 3267 |
This theorem is referenced by: csbcom 3782 csbab 3796 mpt2xopovel 6963 fi1uzind 12647 wrd2ind 12829 elmptrab 20835 sbccom2 32358 sbcrot3 35628 csbabgOLD 37205 |
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