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Mirrors > Home > ILE Home > Th. List > pm5.32ri | GIF version |
Description: Distribution of implication over biconditional (inference rule). (Contributed by NM, 12-Mar-1995.) |
Ref | Expression |
---|---|
pm5.32i.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
pm5.32ri | ⊢ ((𝜓 ∧ 𝜑) ↔ (𝜒 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.32i.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | pm5.32i 441 | . 2 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)) |
3 | ancom 262 | . 2 ⊢ ((𝜓 ∧ 𝜑) ↔ (𝜑 ∧ 𝜓)) | |
4 | ancom 262 | . 2 ⊢ ((𝜒 ∧ 𝜑) ↔ (𝜑 ∧ 𝜒)) | |
5 | 2, 3, 4 | 3bitr4i 210 | 1 ⊢ ((𝜓 ∧ 𝜑) ↔ (𝜒 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: anbi1i 445 pm5.36 574 pm5.61 740 oranabs 761 ceqsralt 2626 ceqsrexbv 2726 reuind 2795 rabsn 3459 dfoprab2 5572 xpsnen 6318 nn1suc 8058 isprm2 10499 |
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