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Mirrors > Home > ILE Home > Th. List > bimsc1 | GIF version |
Description: Removal of conjunct from one side of an equivalence. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
bimsc1 | ⊢ (((𝜑 → 𝜓) ∧ (𝜒 ↔ (𝜓 ∧ 𝜑))) → (𝜒 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 103 | . . . 4 ⊢ ((𝜓 ∧ 𝜑) → 𝜑) | |
2 | ancr 304 | . . . 4 ⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜓 ∧ 𝜑))) | |
3 | 1, 2 | impbid2 131 | . . 3 ⊢ ((𝜑 → 𝜓) → ((𝜓 ∧ 𝜑) ↔ 𝜑)) |
4 | 3 | bibi2d 221 | . 2 ⊢ ((𝜑 → 𝜓) → ((𝜒 ↔ (𝜓 ∧ 𝜑)) ↔ (𝜒 ↔ 𝜑))) |
5 | 4 | biimpa 280 | 1 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 ↔ (𝜓 ∧ 𝜑))) → (𝜒 ↔ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: bm1.3ii 3878 bdbm1.3ii 10011 |
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