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Mirrors > Home > MPE Home > Th. List > 2ax6elem | Structured version Visualization version Unicode version |
Description: We can always find values matching and , as long as they are represented by distinct variables. This theorem merges two ax6e 2250 instances and into a common expression. Alan Sare contributed a variant of this theorem with distinct variable conditions before, see ax6e2nd 38774. (Contributed by Wolf Lammen, 27-Sep-2018.) |
Ref | Expression |
---|---|
2ax6elem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax6e 2250 | . . . 4 | |
2 | nfnae 2318 | . . . . . 6 | |
3 | nfnae 2318 | . . . . . 6 | |
4 | 2, 3 | nfan 1828 | . . . . 5 |
5 | nfeqf 2301 | . . . . . 6 | |
6 | pm3.21 464 | . . . . . 6 | |
7 | 5, 6 | spimed 2255 | . . . . 5 |
8 | 4, 7 | eximd 2085 | . . . 4 |
9 | 1, 8 | mpi 20 | . . 3 |
10 | 9 | ex 450 | . 2 |
11 | ax6e 2250 | . . 3 | |
12 | nfae 2316 | . . . 4 | |
13 | equvini 2346 | . . . . 5 | |
14 | equtrr 1949 | . . . . . . 7 | |
15 | 14 | anim1d 588 | . . . . . 6 |
16 | 15 | aleximi 1759 | . . . . 5 |
17 | 13, 16 | syl5 34 | . . . 4 |
18 | 12, 17 | eximd 2085 | . . 3 |
19 | 11, 18 | mpi 20 | . 2 |
20 | 10, 19 | pm2.61d2 172 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 384 wal 1481 wex 1704 |
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-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 |
This theorem is referenced by: 2ax6e 2450 |
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