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Mirrors > Home > ILE Home > Th. List > 3eqtr4ri | GIF version |
Description: An inference from three chained equalities. (Contributed by NM, 2-Sep-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
3eqtr4i.1 | ⊢ 𝐴 = 𝐵 |
3eqtr4i.2 | ⊢ 𝐶 = 𝐴 |
3eqtr4i.3 | ⊢ 𝐷 = 𝐵 |
Ref | Expression |
---|---|
3eqtr4ri | ⊢ 𝐷 = 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3eqtr4i.3 | . . 3 ⊢ 𝐷 = 𝐵 | |
2 | 3eqtr4i.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
3 | 1, 2 | eqtr4i 2141 | . 2 ⊢ 𝐷 = 𝐴 |
4 | 3eqtr4i.2 | . 2 ⊢ 𝐶 = 𝐴 | |
5 | 3, 4 | eqtr4i 2141 | 1 ⊢ 𝐷 = 𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1316 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1408 ax-gen 1410 ax-4 1472 ax-17 1491 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-cleq 2110 |
This theorem is referenced by: cbvreucsf 3034 dfif6 3446 qdass 3590 tpidm12 3592 unipr 3720 dfdm4 4701 dmun 4716 resres 4801 inres 4806 resdifcom 4807 resiun1 4808 imainrect 4954 coundi 5010 coundir 5011 funopg 5127 offres 6001 mpomptsx 6063 cnvoprab 6099 snec 6458 halfpm6th 8908 numsucc 9189 decbin2 9290 fsumadd 11143 fsum2d 11172 znnen 11838 |
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