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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 2152 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) | |
4 | 1, 2, 3 | mp2an 422 | 1 ⊢ (𝐴 = 𝐶 ↔ 𝐵 = 𝐷) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1331 |
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 1423 ax-gen 1425 ax-4 1487 ax-17 1506 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-cleq 2132 |
This theorem is referenced by: rabbi 2608 sbceqg 3018 preqr2g 3694 preqr2 3696 otth 4164 rncoeq 4812 eqfnov 5877 mpo2eqb 5880 f1o2ndf1 6125 ecopovsym 6525 sq11i 10382 pwle2 13193 |
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