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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege53a | Structured version Visualization version GIF version |
Description: Lemma for frege55a 37182. Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege53a | ⊢ (if-(𝜑, 𝜃, 𝜒) → ((𝜑 ↔ 𝜓) → if-(𝜓, 𝜃, 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-frege52a 37171 | . 2 ⊢ ((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜃, 𝜒) → if-(𝜓, 𝜃, 𝜒))) | |
2 | ax-frege8 37123 | . 2 ⊢ (((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜃, 𝜒) → if-(𝜓, 𝜃, 𝜒))) → (if-(𝜑, 𝜃, 𝜒) → ((𝜑 ↔ 𝜓) → if-(𝜓, 𝜃, 𝜒)))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (if-(𝜑, 𝜃, 𝜒) → ((𝜑 ↔ 𝜓) → if-(𝜓, 𝜃, 𝜒))) |
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-frege8 37123 ax-frege52a 37171 |
This theorem is referenced by: frege55a 37182 |
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