![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > alanimi | Structured version Visualization version GIF version |
Description: Variant of al2imi 1892 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.) |
Ref | Expression |
---|---|
alanimi.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Ref | Expression |
---|---|
alanimi | ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alanimi.1 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | |
2 | 1 | ex 449 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) |
3 | 2 | al2imi 1892 | . 2 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) |
4 | 3 | imp 444 | 1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 |
This theorem depends on definitions: df-bi 197 df-an 385 |
This theorem is referenced by: 19.26 1947 alsyl 1969 ax13 2394 nfeqf 2446 bm1.1 2745 vtoclgft 3394 vtoclgftOLD 3395 euind 3534 reuind 3552 sbeqalb 3629 bm1.3ii 4936 trin2 5677 bj-cbv3ta 33038 mpt2bi123f 34302 mptbi12f 34306 cotrintab 38441 albitr 39082 2alanimi 39091 |
Copyright terms: Public domain | W3C validator |