New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > trd | Unicode version |
Description: Transitivity law in natural deduction form. (Contributed by SF, 20-Feb-2015.) |
Ref | Expression |
---|---|
trd.1 | Trans |
trd.2 | |
trd.3 | |
trd.4 | |
trd.5 | |
trd.6 |
Ref | Expression |
---|---|
trd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trd.5 | . 2 | |
2 | trd.6 | . 2 | |
3 | trd.1 | . . . 4 Trans | |
4 | brex 4689 | . . . . . 6 Trans | |
5 | breq 4641 | . . . . . . . . . . 11 | |
6 | breq 4641 | . . . . . . . . . . 11 | |
7 | 5, 6 | anbi12d 691 | . . . . . . . . . 10 |
8 | breq 4641 | . . . . . . . . . 10 | |
9 | 7, 8 | imbi12d 311 | . . . . . . . . 9 |
10 | 9 | ralbidv 2634 | . . . . . . . 8 |
11 | 10 | 2ralbidv 2656 | . . . . . . 7 |
12 | raleq 2807 | . . . . . . . . 9 | |
13 | 12 | raleqbi1dv 2815 | . . . . . . . 8 |
14 | 13 | raleqbi1dv 2815 | . . . . . . 7 |
15 | df-trans 5899 | . . . . . . 7 Trans | |
16 | 11, 14, 15 | brabg 4706 | . . . . . 6 Trans |
17 | 4, 16 | syl 15 | . . . . 5 Trans Trans |
18 | 17 | ibi 232 | . . . 4 Trans |
19 | 3, 18 | syl 15 | . . 3 |
20 | trd.2 | . . . 4 | |
21 | trd.3 | . . . 4 | |
22 | trd.4 | . . . 4 | |
23 | breq1 4642 | . . . . . . 7 | |
24 | 23 | anbi1d 685 | . . . . . 6 |
25 | breq1 4642 | . . . . . 6 | |
26 | 24, 25 | imbi12d 311 | . . . . 5 |
27 | breq2 4643 | . . . . . . 7 | |
28 | breq1 4642 | . . . . . . 7 | |
29 | 27, 28 | anbi12d 691 | . . . . . 6 |
30 | 29 | imbi1d 308 | . . . . 5 |
31 | breq2 4643 | . . . . . . 7 | |
32 | 31 | anbi2d 684 | . . . . . 6 |
33 | breq2 4643 | . . . . . 6 | |
34 | 32, 33 | imbi12d 311 | . . . . 5 |
35 | 26, 30, 34 | rspc3v 2964 | . . . 4 |
36 | 20, 21, 22, 35 | syl3anc 1182 | . . 3 |
37 | 19, 36 | mpd 14 | . 2 |
38 | 1, 2, 37 | mp2and 660 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wa 358 wceq 1642 wcel 1710 wral 2614 cvv 2859 class class class wbr 4639 Trans ctrans 5888 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-trans 5899 |
This theorem is referenced by: ertr 5954 ertrd 5955 |
Copyright terms: Public domain | W3C validator |