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Mirrors > Home > ILE Home > Th. List > 3eqtr4a | GIF version |
Description: A chained equality inference, useful for converting to definitions. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
3eqtr4a.1 | ⊢ 𝐴 = 𝐵 |
3eqtr4a.2 | ⊢ (𝜑 → 𝐶 = 𝐴) |
3eqtr4a.3 | ⊢ (𝜑 → 𝐷 = 𝐵) |
Ref | Expression |
---|---|
3eqtr4a | ⊢ (𝜑 → 𝐶 = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3eqtr4a.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐴) | |
2 | 3eqtr4a.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
3 | 1, 2 | syl6eq 2129 | . 2 ⊢ (𝜑 → 𝐶 = 𝐵) |
4 | 3eqtr4a.3 | . 2 ⊢ (𝜑 → 𝐷 = 𝐵) | |
5 | 3, 4 | eqtr4d 2116 | 1 ⊢ (𝜑 → 𝐶 = 𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-4 1440 ax-17 1459 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-cleq 2074 |
This theorem is referenced by: uniintsnr 3672 fndmdifcom 5294 offres 5782 1stval2 5802 2ndval2 5803 ecovcom 6236 ecovass 6238 ecovdi 6240 zeo 8452 xnegneg 8900 fzsuc2 9096 expnegap0 9484 facp1 9657 bcpasc 9693 absexp 9965 gcdcom 10365 gcd0id 10370 dfgcd3 10399 gcdass 10404 lcmcom 10446 lcmneg 10456 lcmass 10467 sqrt2irrlem 10540 |
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