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Mirrors > Home > MPE Home > Th. List > pm4.38 | Structured version Visualization version GIF version |
Description: Theorem *4.38 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.) |
Ref | Expression |
---|---|
pm4.38 | ⊢ (((𝜑 ↔ 𝜒) ∧ (𝜓 ↔ 𝜃)) → ((𝜑 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 486 | . 2 ⊢ (((𝜑 ↔ 𝜒) ∧ (𝜓 ↔ 𝜃)) → (𝜑 ↔ 𝜒)) | |
2 | simpr 488 | . 2 ⊢ (((𝜑 ↔ 𝜒) ∧ (𝜓 ↔ 𝜃)) → (𝜓 ↔ 𝜃)) | |
3 | 1, 2 | anbi12d 634 | 1 ⊢ (((𝜑 ↔ 𝜒) ∧ (𝜓 ↔ 𝜃)) → ((𝜑 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 |
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 210 df-an 400 |
This theorem is referenced by: bi2anan9 639 xpf1o 8722 isprm3 16117 csbingVD 42026 csbxpgVD 42036 csbunigVD 42040 ichan 44425 |
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