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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege57a | Structured version Visualization version GIF version |
Description: Analogue of frege57aid 42394. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege57a | ⊢ ((𝜑 ↔ 𝜓) → (if-(𝜓, 𝜒, 𝜃) → if-(𝜑, 𝜒, 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-frege52a 42379 | . 2 ⊢ ((𝜓 ↔ 𝜑) → (if-(𝜓, 𝜒, 𝜃) → if-(𝜑, 𝜒, 𝜃))) | |
2 | frege56a 42393 | . 2 ⊢ (((𝜓 ↔ 𝜑) → (if-(𝜓, 𝜒, 𝜃) → if-(𝜑, 𝜒, 𝜃))) → ((𝜑 ↔ 𝜓) → (if-(𝜓, 𝜒, 𝜃) → if-(𝜑, 𝜒, 𝜃)))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ((𝜑 ↔ 𝜓) → (if-(𝜓, 𝜒, 𝜃) → if-(𝜑, 𝜒, 𝜃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 if-wif 1061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-frege1 42312 ax-frege2 42313 ax-frege8 42331 ax-frege28 42352 ax-frege52a 42379 ax-frege54a 42384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ifp 1062 |
This theorem is referenced by: (None) |
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