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Mirrors > Home > ILE Home > Th. List > bicom1 | GIF version |
Description: Commutative law for equivalence. (Contributed by Wolf Lammen, 10-Nov-2012.) |
Ref | Expression |
---|---|
bicom1 | ⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biimpr 130 | . 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑)) | |
2 | biimp 118 | . 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)) | |
3 | 1, 2 | impbid 129 | 1 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: bicomi 132 bicom 140 pm5.21ndd 705 cbvexdh 1926 elabgf2 14392 |
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