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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 2195 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶)) | |
2 | eqeq2 2198 | . 2 ⊢ (𝐶 = 𝐷 → (𝐵 = 𝐶 ↔ 𝐵 = 𝐷)) | |
3 | 1, 2 | sylan9bb 462 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1363 |
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-5 1457 ax-gen 1459 ax-4 1520 ax-17 1536 ax-ext 2170 |
This theorem depends on definitions: df-bi 117 df-cleq 2181 |
This theorem is referenced by: eqeq12i 2202 eqeq12d 2203 eqeqan12d 2204 funopg 5264 tfri3 6385 th3qlem1 6654 xpdom2 6848 difinfsnlem 7115 difinfsn 7116 xrlttri3 9814 bcn1 10755 summodc 11408 prodmodc 11603 ringinvnz1ne0 13361 |
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