![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > anim12d1 | Structured version Visualization version GIF version |
Description: Variant of anim12d 608 where the second implication does not depend on the antecedent. (Contributed by Rodolfo Medina, 12-Oct-2010.) |
Ref | Expression |
---|---|
anim12d1.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
anim12d1.2 | ⊢ (𝜃 → 𝜏) |
Ref | Expression |
---|---|
anim12d1 | ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → (𝜒 ∧ 𝜏))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anim12d1.1 | . 2 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | anim12d1.2 | . . 3 ⊢ (𝜃 → 𝜏) | |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → (𝜃 → 𝜏)) |
4 | 1, 3 | anim12d 608 | 1 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → (𝜒 ∧ 𝜏))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 |
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 206 df-an 396 |
This theorem is referenced by: fun 6747 frrlem13 8284 alephord 10072 grudomon 10814 xrsupexmnf 13290 xrinfmexpnf 13291 joinfval 18338 meetfval 18352 cnpresti 23147 1stcrest 23312 upgrwlkdvdelem 29502 |
Copyright terms: Public domain | W3C validator |