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Mirrors > Home > MPE Home > Th. List > Mathboxes > equid1 | Structured version Visualization version Unicode version |
Description: Proof of equid 1939 from our older axioms. This is often an axiom of equality in textbook systems, but we don't need it as an axiom since it can be proved from our other axioms (although the proof, as you can see below, is not as obvious as you might think). This proof uses only axioms without distinct variable conditions and requires no dummy variables. A simpler proof, similar to Tarski's, is possible if we make use of ax-5 1839; see the proof of equid 1939. See equid1ALT 34210 for an alternate proof. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
equid1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-c4 34169 | . . . 4 | |
2 | ax-c5 34168 | . . . . 5 | |
3 | ax-c9 34175 | . . . . 5 | |
4 | 2, 2, 3 | sylc 65 | . . . 4 |
5 | 1, 4 | mpg 1724 | . . 3 |
6 | ax-c10 34171 | . . 3 | |
7 | 5, 6 | syl 17 | . 2 |
8 | ax-c7 34170 | . 2 | |
9 | 7, 8 | pm2.61i 176 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wal 1481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-c5 34168 ax-c4 34169 ax-c7 34170 ax-c10 34171 ax-c9 34175 |
This theorem is referenced by: equcomi1 34185 |
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