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Mirrors > Home > MPE Home > Th. List > sylan9eq | Structured version Visualization version GIF version |
Description: An equality transitivity deduction. (Contributed by NM, 8-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
sylan9eq.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
sylan9eq.2 | ⊢ (𝜓 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
sylan9eq | ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylan9eq.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | sylan9eq.2 | . 2 ⊢ (𝜓 → 𝐵 = 𝐶) | |
3 | eqtr 2761 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐶) | |
4 | 1, 2, 3 | syl2an 596 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) |
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