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Mirrors > Home > MPE Home > Th. List > ifpbi23d | Structured version Visualization version GIF version |
Description: Equivalence deduction for conditional operator for propositions. Convenience theorem for a frequent case. (Contributed by Wolf Lammen, 28-Apr-2024.) |
Ref | Expression |
---|---|
ifpbi23d.1 | ⊢ (𝜑 → (𝜒 ↔ 𝜂)) |
ifpbi23d.2 | ⊢ (𝜑 → (𝜃 ↔ 𝜁)) |
Ref | Expression |
---|---|
ifpbi23d | ⊢ (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜓, 𝜂, 𝜁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biidd 265 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜓)) | |
2 | ifpbi23d.1 | . 2 ⊢ (𝜑 → (𝜒 ↔ 𝜂)) | |
3 | ifpbi23d.2 | . 2 ⊢ (𝜑 → (𝜃 ↔ 𝜁)) | |
4 | 1, 2, 3 | ifpbi123d 1080 | 1 ⊢ (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜓, 𝜂, 𝜁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 if-wif 1063 |
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 210 df-an 400 df-or 848 df-ifp 1064 |
This theorem is referenced by: wksfval 27651 subgrwlk 32761 satfv1fvfmla1 33052 bj-ififc 34449 wl-df-3xor 35325 wl-df3maxtru1 35349 |
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