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Mirrors > Home > MPE Home > Th. List > 3eqtr4a | Structured version Visualization version 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 | eqtrdi 2796 | . 2 ⊢ (𝜑 → 𝐶 = 𝐵) |
4 | 3eqtr4a.3 | . 2 ⊢ (𝜑 → 𝐷 = 𝐵) | |
5 | 3, 4 | eqtr4d 2783 | 1 ⊢ (𝜑 → 𝐶 = 𝐷) |
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