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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 2190 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) | |
4 | 1, 2, 3 | mp2an 426 | 1 ⊢ (𝐴 = 𝐶 ↔ 𝐵 = 𝐷) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1353 |
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 1447 ax-gen 1449 ax-4 1510 ax-17 1526 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-cleq 2170 |
This theorem is referenced by: rabbi 2655 sbceqg 3074 preqr2g 3768 preqr2 3770 otth 4243 rncoeq 4901 eqfnov 5981 mpo2eqb 5984 f1o2ndf1 6229 ecopovsym 6631 sq11i 10610 pwle2 14751 |
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