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Mirrors > Home > ILE Home > Th. List > 3eqtr3g | GIF version |
Description: A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.) |
Ref | Expression |
---|---|
3eqtr3g.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
3eqtr3g.2 | ⊢ 𝐴 = 𝐶 |
3eqtr3g.3 | ⊢ 𝐵 = 𝐷 |
Ref | Expression |
---|---|
3eqtr3g | ⊢ (𝜑 → 𝐶 = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3eqtr3g.2 | . . 3 ⊢ 𝐴 = 𝐶 | |
2 | 3eqtr3g.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 1, 2 | syl5eqr 2187 | . 2 ⊢ (𝜑 → 𝐶 = 𝐵) |
4 | 3eqtr3g.3 | . 2 ⊢ 𝐵 = 𝐷 | |
5 | 3, 4 | eqtrdi 2189 | 1 ⊢ (𝜑 → 𝐶 = 𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 |
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 1424 ax-gen 1426 ax-4 1488 ax-17 1507 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-cleq 2133 |
This theorem is referenced by: csbnest1g 3060 disjdif2 3446 dfopg 3711 xpid11 4770 sqxpeq0 4970 cores2 5059 funcoeqres 5406 dftpos2 6166 ine0 8180 fisumcom2 11239 fisum0diag2 11248 mertenslemi1 11336 setsslnid 12049 dvmptccn 12887 nninffeq 13391 |
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