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Mirrors > Home > ILE Home > Th. List > oteq123d | GIF version |
Description: Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
oteq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
oteq123d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
oteq123d.3 | ⊢ (𝜑 → 𝐸 = 𝐹) |
Ref | Expression |
---|---|
oteq123d | ⊢ (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oteq1d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | oteq1d 3805 | . 2 ⊢ (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐶, 𝐸⟩) |
3 | oteq123d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | oteq2d 3806 | . 2 ⊢ (𝜑 → ⟨𝐵, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐸⟩) |
5 | oteq123d.3 | . . 3 ⊢ (𝜑 → 𝐸 = 𝐹) | |
6 | 5 | oteq3d 3807 | . 2 ⊢ (𝜑 → ⟨𝐵, 𝐷, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩) |
7 | 2, 4, 6 | 3eqtrd 2226 | 1 ⊢ (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ⟨cotp 3611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-un 3148 df-sn 3613 df-pr 3614 df-op 3616 df-ot 3617 |
This theorem is referenced by: (None) |
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