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Mirrors > Home > MPE Home > Th. List > alanimi | Structured version Visualization version GIF version |
Description: Variant of al2imi 1733 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 1733 | . 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 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 |
This theorem depends on definitions: df-bi 196 df-an 385 |
This theorem is referenced by: 19.26 1786 alsyl 1811 ax13 2237 nfeqf 2289 bm1.1 2595 vtoclgft 3227 vtoclgftOLD 3228 euind 3360 reuind 3378 sbeqalb 3455 bm1.3ii 4712 trin2 5438 bj-cbv3ta 31897 mpt2bi123f 33141 mptbi12f 33145 cotrintab 36940 albitr 37584 2alanimi 37593 |
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