New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > dedth | Unicode version |
Description: Weak deduction theorem that eliminates a hypothesis , making it become an antecedent. We assume that a proof exists for when the class variable is replaced with a specific class . The hypothesis should be assigned to the inference, and the inference's hypothesis eliminated with elimhyp 3710. If the inference has other hypotheses with class variable , these can be kept by assigning keephyp 3716 to them. For more information, see the Deduction Theorem https://us.metamath.org/mpeuni/mmdeduction.html 3716. (Contributed by NM, 15-May-1999.) |
Ref | Expression |
---|---|
dedth.1 | |
dedth.2 |
Ref | Expression |
---|---|
dedth |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dedth.2 | . 2 | |
2 | iftrue 3668 | . . . 4 | |
3 | 2 | eqcomd 2358 | . . 3 |
4 | dedth.1 | . . 3 | |
5 | 3, 4 | syl 15 | . 2 |
6 | 1, 5 | mpbiri 224 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wceq 1642 cif 3662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-if 3663 |
This theorem is referenced by: dedth2h 3704 dedth3h 3705 |
Copyright terms: Public domain | W3C validator |