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Mirrors > Home > ILE Home > Th. List > pm5.32rd | GIF version |
Description: Distribution of implication over biconditional (deduction rule). (Contributed by NM, 25-Dec-2004.) |
Ref | Expression |
---|---|
pm5.32d.1 | ⊢ (φ → (ψ → (χ ↔ θ))) |
Ref | Expression |
---|---|
pm5.32rd | ⊢ (φ → ((χ ∧ ψ) ↔ (θ ∧ ψ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.32d.1 | . . 3 ⊢ (φ → (ψ → (χ ↔ θ))) | |
2 | 1 | pm5.32d 423 | . 2 ⊢ (φ → ((ψ ∧ χ) ↔ (ψ ∧ θ))) |
3 | ancom 253 | . 2 ⊢ ((χ ∧ ψ) ↔ (ψ ∧ χ)) | |
4 | ancom 253 | . 2 ⊢ ((θ ∧ ψ) ↔ (ψ ∧ θ)) | |
5 | 2, 3, 4 | 3bitr4g 212 | 1 ⊢ (φ → ((χ ∧ ψ) ↔ (θ ∧ ψ))) |
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: anbi1d 438 pm5.71dc 867 1idprl 6566 1idpru 6567 |
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