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Mirrors > Home > ILE Home > Th. List > 3anbi3d | GIF version |
Description: Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.) |
Ref | Expression |
---|---|
3anbi1d.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
3anbi3d | ⊢ (𝜑 → ((𝜃 ∧ 𝜏 ∧ 𝜓) ↔ (𝜃 ∧ 𝜏 ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biidd 171 | . 2 ⊢ (𝜑 → (𝜃 ↔ 𝜃)) | |
2 | 3anbi1d.1 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | 3anbi13d 1292 | 1 ⊢ (𝜑 → ((𝜃 ∧ 𝜏 ∧ 𝜓) ↔ (𝜃 ∧ 𝜏 ∧ 𝜒))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∧ w3a 962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 df-3an 964 |
This theorem is referenced by: ceqsex3v 2723 ceqsex4v 2724 ceqsex8v 2726 vtocl3gaf 2750 mob 2861 ordsoexmid 4472 tfr1onlemaccex 6238 tfrcllemaccex 6251 fseq1m1p1 9868 summodc 11145 fsum3 11149 divalglemnn 11604 divalglemeunn 11607 divalglemex 11608 divalglemeuneg 11609 |
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