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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 | pm5.501 369 | . 2 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ↔ 𝜓))) | |
2 | 1 | biimpa 479 | 1 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 |
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 209 df-an 399 |
This theorem is referenced by: pm5.35 823 impimprbi 826 ssconb 4114 raaan 4460 raaanv 4461 raaan2 4464 suppimacnvss 7840 mdsymi 30188 tsbi1 35426 abnotbtaxb 43171 elprneb 43284 |
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