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Mirrors > Home > ILE Home > Th. List > ineq12d | GIF version |
Description: Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
ineq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
ineq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
ineq12d | ⊢ (𝜑 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | ineq12d.2 | . 2 ⊢ (𝜑 → 𝐶 = 𝐷) | |
3 | ineq12 3267 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) | |
4 | 1, 2, 3 | syl2anc 408 | 1 ⊢ (𝜑 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∩ cin 3065 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-in 3072 |
This theorem is referenced by: csbing 3278 funprg 5168 funtpg 5169 offval 5982 ofrfval 5983 undifdc 6805 djudom 6971 ressid2 12007 ressval2 12008 |
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