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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege54cor0a | Structured version Visualization version GIF version |
Description: Synonym for logical equivalence. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege54cor0a | ⊢ ((𝜓 ↔ 𝜑) ↔ if-(𝜓, 𝜑, ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-frege28 37144 | . . . 4 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑)) | |
2 | 1 | anim2i 591 | . . 3 ⊢ (((𝜓 → 𝜑) ∧ (𝜑 → 𝜓)) → ((𝜓 → 𝜑) ∧ (¬ 𝜓 → ¬ 𝜑))) |
3 | con4 111 | . . . 4 ⊢ ((¬ 𝜓 → ¬ 𝜑) → (𝜑 → 𝜓)) | |
4 | 3 | anim2i 591 | . . 3 ⊢ (((𝜓 → 𝜑) ∧ (¬ 𝜓 → ¬ 𝜑)) → ((𝜓 → 𝜑) ∧ (𝜑 → 𝜓))) |
5 | 2, 4 | impbii 198 | . 2 ⊢ (((𝜓 → 𝜑) ∧ (𝜑 → 𝜓)) ↔ ((𝜓 → 𝜑) ∧ (¬ 𝜓 → ¬ 𝜑))) |
6 | dfbi2 658 | . 2 ⊢ ((𝜓 ↔ 𝜑) ↔ ((𝜓 → 𝜑) ∧ (𝜑 → 𝜓))) | |
7 | dfifp2 1008 | . 2 ⊢ (if-(𝜓, 𝜑, ¬ 𝜑) ↔ ((𝜓 → 𝜑) ∧ (¬ 𝜓 → ¬ 𝜑))) | |
8 | 5, 6, 7 | 3bitr4i 291 | 1 ⊢ ((𝜓 ↔ 𝜑) ↔ if-(𝜓, 𝜑, ¬ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 if-wif 1006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-frege28 37144 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 |
This theorem is referenced by: frege54cor1a 37178 frege55lem1a 37180 frege55lem2a 37181 |
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