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Mirrors > Home > MPE Home > Th. List > Mathboxes > ifpbi23 | Structured version Visualization version GIF version |
Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.) |
Ref | Expression |
---|---|
ifpbi23 | ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (if-(𝜏, 𝜑, 𝜒) ↔ if-(𝜏, 𝜓, 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 484 | . 2 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (𝜑 ↔ 𝜓)) | |
2 | simpr 486 | . 2 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (𝜒 ↔ 𝜃)) | |
3 | 1, 2 | ifpbi23d 1081 | 1 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (if-(𝜏, 𝜑, 𝜒) ↔ if-(𝜏, 𝜓, 𝜃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 if-wif 1062 |
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 206 df-an 398 df-or 847 df-ifp 1063 |
This theorem is referenced by: ifpnot23d 41831 ifpdfxor 41833 ifpananb 41852 ifpnannanb 41853 ifpxorxorb 41857 |
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