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| Mirrors > Home > MPE Home > Th. List > ala1 | Structured version Visualization version GIF version | ||
| Description: Add an antecedent in a universally quantified formula. (Contributed by BJ, 6-Oct-2018.) | 
| Ref | Expression | 
|---|---|
| ala1 | ⊢ (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ax-1 6 | . 2 ⊢ (𝜑 → (𝜓 → 𝜑)) | |
| 2 | 1 | alimi 1811 | 1 ⊢ (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1538 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-gen 1795 ax-4 1809 | 
| This theorem is referenced by: 19.38 1839 stdpc4 2068 ax12dgen 2134 ax12 2428 sb4a 2485 alral 3075 hbimtg 35807 bj-axdd2 36593 bj-ax12v3ALT 36687 bj-equsal1t 36823 | 
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