Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 19.3v | Structured version Visualization version GIF version |
Description: Version of 19.3 2195 with a disjoint variable condition, requiring fewer axioms. Any formula can be universally quantified using a variable which it does not contain. See also 19.9v 1987. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 2011. (Revised by Wolf Lammen, 4-Dec-2017.) (Proof shortened by Wolf Lammen, 20-Oct-2023.) |
Ref | Expression |
---|---|
19.3v | ⊢ (∀𝑥𝜑 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spvw 1984 | . 2 ⊢ (∀𝑥𝜑 → 𝜑) | |
2 | ax-5 1913 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | 1, 2 | impbii 208 | 1 ⊢ (∀𝑥𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∀wal 1537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 |
This theorem depends on definitions: df-bi 206 df-ex 1783 |
This theorem is referenced by: 19.27v 1993 19.28v 1994 19.37v 1995 axrep1 5210 axrep6 5216 kmlem14 9919 zfcndrep 10370 zfcndpow 10372 zfcndac 10375 lfuhgr3 33081 bj-snsetex 35153 iooelexlt 35533 dford4 40851 relexp0eq 41309 |
Copyright terms: Public domain | W3C validator |