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Mirrors > Home > MPE Home > Th. List > pm5.1 | Structured version Visualization version GIF version |
Description: Two propositions are equivalent if they are both true. Theorem *5.1 of [WhiteheadRussell] p. 123. (Contributed by NM, 21-May-1994.) |
Ref | Expression |
---|---|
pm5.1 | ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.501 370 | . 2 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ↔ 𝜓))) | |
2 | 1 | biimpa 480 | 1 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 |
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 |
This theorem is referenced by: pm5.35 824 impimprbi 827 ssconb 4065 raaan 4418 raaanv 4419 raaan2 4422 suppimacnvss 7823 mdsymi 30194 tsbi1 35571 abnotbtaxb 43508 elprneb 43621 |
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