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Mirrors > Home > NFE Home > Th. List > mpbir2and | GIF version |
Description: Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) |
Ref | Expression |
---|---|
mpbir2and.1 | ⊢ (φ → χ) |
mpbir2and.2 | ⊢ (φ → θ) |
mpbir2and.3 | ⊢ (φ → (ψ ↔ (χ ∧ θ))) |
Ref | Expression |
---|---|
mpbir2and | ⊢ (φ → ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpbir2and.1 | . . 3 ⊢ (φ → χ) | |
2 | mpbir2and.2 | . . 3 ⊢ (φ → θ) | |
3 | 1, 2 | jca 518 | . 2 ⊢ (φ → (χ ∧ θ)) |
4 | mpbir2and.3 | . 2 ⊢ (φ → (ψ ↔ (χ ∧ θ))) | |
5 | 3, 4 | mpbird 223 | 1 ⊢ (φ → ψ) |
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: fvopab4t 5385 |
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