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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege56a | Structured version Visualization version GIF version |
Description: Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege56a | ⊢ (((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃))) → ((𝜓 ↔ 𝜑) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege55cor1a 37183 | . 2 ⊢ ((𝜓 ↔ 𝜑) → (𝜑 ↔ 𝜓)) | |
2 | frege9 37126 | . 2 ⊢ (((𝜓 ↔ 𝜑) → (𝜑 ↔ 𝜓)) → (((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃))) → ((𝜓 ↔ 𝜑) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃))))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃))) → ((𝜓 ↔ 𝜑) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 if-wif 1006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-frege1 37104 ax-frege2 37105 ax-frege8 37123 ax-frege28 37144 ax-frege52a 37171 ax-frege54a 37176 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 |
This theorem is referenced by: frege57a 37187 |
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