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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbccomlem 3508 | . . . 4 | |
2 | sbccomlem 3508 | . . . . . . 7 | |
3 | 2 | sbcbii 3491 | . . . . . 6 |
4 | sbccomlem 3508 | . . . . . 6 | |
5 | 3, 4 | bitri 264 | . . . . 5 |
6 | 5 | sbcbii 3491 | . . . 4 |
7 | sbccomlem 3508 | . . . . 5 | |
8 | 7 | sbcbii 3491 | . . . 4 |
9 | 1, 6, 8 | 3bitr3i 290 | . . 3 |
10 | sbcco 3458 | . . 3 | |
11 | sbcco 3458 | . . 3 | |
12 | 9, 10, 11 | 3bitr3i 290 | . 2 |
13 | sbcco 3458 | . . 3 | |
14 | 13 | sbcbii 3491 | . 2 |
15 | sbcco 3458 | . . 3 | |
16 | 15 | sbcbii 3491 | . 2 |
17 | 12, 14, 16 | 3bitr3i 290 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wsbc 3435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-v 3202 df-sbc 3436 |
This theorem is referenced by: csbcom 3994 csbab 4008 mpt2xopovel 7346 fi1uzind 13279 fi1uzindOLD 13285 wrd2ind 13477 elmptrab 21630 sbccom2 33930 sbcrot3 37355 csbabgOLD 39050 |
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