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Mirrors > Home > NFE Home > Th. List > baibd | GIF version |
Description: Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
baibd.1 | ⊢ (φ → (ψ ↔ (χ ∧ θ))) |
Ref | Expression |
---|---|
baibd | ⊢ ((φ ∧ χ) → (ψ ↔ θ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | baibd.1 | . 2 ⊢ (φ → (ψ ↔ (χ ∧ θ))) | |
2 | ibar 490 | . . 3 ⊢ (χ → (θ ↔ (χ ∧ θ))) | |
3 | 2 | bicomd 192 | . 2 ⊢ (χ → ((χ ∧ θ) ↔ θ)) |
4 | 1, 3 | sylan9bb 680 | 1 ⊢ ((φ ∧ χ) → (ψ ↔ θ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 |
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 177 df-an 360 |
This theorem is referenced by: (None) |
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