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Mirrors > Home > ILE Home > Th. List > eqeq12 | GIF version |
Description: Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.) |
Ref | Expression |
---|---|
eqeq12 | ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2177 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶)) | |
2 | eqeq2 2180 | . 2 ⊢ (𝐶 = 𝐷 → (𝐵 = 𝐶 ↔ 𝐵 = 𝐷)) | |
3 | 1, 2 | sylan9bb 459 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1348 |
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 1440 ax-gen 1442 ax-4 1503 ax-17 1519 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-cleq 2163 |
This theorem is referenced by: eqeq12i 2184 eqeq12d 2185 eqeqan12d 2186 funopg 5230 tfri3 6344 th3qlem1 6613 xpdom2 6807 difinfsnlem 7074 difinfsn 7075 xrlttri3 9747 bcn1 10685 summodc 11339 prodmodc 11534 |
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