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Mirrors > Home > MPE Home > Th. List > Mathboxes > 3orbi123 | Structured version Visualization version GIF version |
Description: pm4.39 962 with a 3-conjunct antecedent. This proof is 3orbi123VD 40001 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
3orbi123 | ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1127 | . 2 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → (𝜑 ↔ 𝜓)) | |
2 | simp2 1128 | . 2 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → (𝜒 ↔ 𝜃)) | |
3 | simp3 1129 | . 2 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → (𝜏 ↔ 𝜂)) | |
4 | 1, 2, 3 | 3orbi123d 1508 | 1 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∨ w3o 1070 ∧ w3a 1071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 |
This theorem is referenced by: sbcoreleleq 39677 sbcoreleleqVD 40010 |
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