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Mirrors > Home > ILE Home > Th. List > alanimi | GIF version |
Description: Variant of al2imi 1451 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 114 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) |
3 | 2 | al2imi 1451 | . 2 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) |
4 | 3 | imp 123 | 1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1440 ax-gen 1442 |
This theorem is referenced by: 19.26 1474 alsyl 1628 vtoclgft 2780 euind 2917 reuind 2935 sbeqalb 3011 bm1.3ii 4108 trin2 5000 bdbm1.3ii 13848 |
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