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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > equid1 | Structured version Visualization version Unicode version |
Description: Proof of equid 1855 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 1758; see the proof of equid 1855. See equid1ALT 32496 for an alternate proof. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
equid1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-c4 32456 |
. . . 4
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2 | ax-c5 32455 |
. . . . 5
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3 | ax-c9 32462 |
. . . . 5
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4 | 2, 2, 3 | sylc 62 |
. . . 4
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5 | 1, 4 | mpg 1671 |
. . 3
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6 | ax-c10 32458 |
. . 3
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7 | 5, 6 | syl 17 |
. 2
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8 | ax-c7 32457 |
. 2
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9 | 7, 8 | pm2.61i 168 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1669 ax-c5 32455 ax-c4 32456 ax-c7 32457 ax-c10 32458 ax-c9 32462 |
This theorem is referenced by: (None) |
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