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Mirrors > Home > ILE Home > Th. List > eqeq12i | GIF version |
Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
eqeq12i.1 | ⊢ 𝐴 = 𝐵 |
eqeq12i.2 | ⊢ 𝐶 = 𝐷 |
Ref | Expression |
---|---|
eqeq12i | ⊢ (𝐴 = 𝐶 ↔ 𝐵 = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq12i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | eqeq12i.2 | . 2 ⊢ 𝐶 = 𝐷 | |
3 | eqeq12 2177 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) | |
4 | 1, 2, 3 | mp2an 423 | 1 ⊢ (𝐴 = 𝐶 ↔ 𝐵 = 𝐷) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1342 |
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 1434 ax-gen 1436 ax-4 1497 ax-17 1513 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-cleq 2157 |
This theorem is referenced by: rabbi 2641 sbceqg 3057 preqr2g 3742 preqr2 3744 otth 4215 rncoeq 4872 eqfnov 5940 mpo2eqb 5943 f1o2ndf1 6188 ecopovsym 6589 sq11i 10535 pwle2 13739 |
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