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Mirrors > Home > MPE Home > Th. List > hbex | Structured version Visualization version GIF version |
Description: If 𝑥 is not free in 𝜑, it is not free in ∃𝑦𝜑. (Contributed by NM, 12-Mar-1993.) Reduce symbol count in nfex 2343, hbex 2344. (Revised by Wolf Lammen, 16-Oct-2021.) |
Ref | Expression |
---|---|
hbex.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
Ref | Expression |
---|---|
hbex | ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbex.1 | . . . 4 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | 1 | nf5i 2150 | . . 3 ⊢ Ⅎ𝑥𝜑 |
3 | 2 | nfex 2343 | . 2 ⊢ Ⅎ𝑥∃𝑦𝜑 |
4 | 3 | nf5ri 2195 | 1 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 ∃wex 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-11 2161 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-or 844 df-ex 1781 df-nf 1785 |
This theorem is referenced by: (None) |
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