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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 2129 | . 2 ⊢ (𝜑 → 𝐵 = 𝐶) |
5 | 1, 4 | eqtr3d 2129 | 1 ⊢ (𝜑 → 𝐷 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1296 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1388 ax-gen 1390 ax-4 1452 ax-17 1471 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-cleq 2088 |
This theorem is referenced by: fcofo 5601 fcof1o 5606 frecabcl 6202 nnaword 6310 enomnilem 6881 fodju0 6890 pn0sr 7414 negeu 7770 add20 8049 2halves 8743 bcnn 10280 bcpasc 10289 resqrexlemover 10558 fsumneg 10994 geolim 11054 geolim2 11055 mertensabs 11080 sincossq 11188 demoivre 11211 eirraplem 11213 gcdid 11404 phiprmpw 11625 ioo2bl 12319 |
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