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Mirrors > Home > ILE Home > Th. List > 3eqtr3rd | GIF version |
Description: A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.) |
Ref | Expression |
---|---|
3eqtr3d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
3eqtr3d.2 | ⊢ (𝜑 → 𝐴 = 𝐶) |
3eqtr3d.3 | ⊢ (𝜑 → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
3eqtr3rd | ⊢ (𝜑 → 𝐷 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3eqtr3d.3 | . 2 ⊢ (𝜑 → 𝐵 = 𝐷) | |
2 | 3eqtr3d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 3eqtr3d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐶) | |
4 | 2, 3 | eqtr3d 2174 | . 2 ⊢ (𝜑 → 𝐵 = 𝐶) |
5 | 1, 4 | eqtr3d 2174 | 1 ⊢ (𝜑 → 𝐷 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 |
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 1423 ax-gen 1425 ax-4 1487 ax-17 1506 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-cleq 2132 |
This theorem is referenced by: fcofo 5685 fcof1o 5690 frecabcl 6296 nnaword 6407 enomnilem 7010 fodju0 7019 pn0sr 7579 negeu 7953 add20 8236 2halves 8949 bcnn 10503 bcpasc 10512 resqrexlemover 10782 fsumneg 11220 geolim 11280 geolim2 11281 mertensabs 11306 sincossq 11455 demoivre 11479 eirraplem 11483 gcdid 11674 gcdmultipled 11681 phiprmpw 11898 ioo2bl 12712 ptolemy 12905 coskpi 12929 |
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