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Mirrors > Home > MPE Home > Th. List > 19.3v | Structured version Visualization version GIF version |
Description: Version of 19.3 2204 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 1993. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 2020. (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 1990 | . 2 ⊢ (∀𝑥𝜑 → 𝜑) | |
2 | ax-5 1917 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | 1, 2 | impbii 212 | 1 ⊢ (∀𝑥𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∀wal 1540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 |
This theorem depends on definitions: df-bi 210 df-ex 1787 |
This theorem is referenced by: spvwOLD 1995 19.27v 2001 19.28v 2002 19.37v 2003 axrep1 5165 axrep6 5171 kmlem14 9675 zfcndrep 10126 zfcndpow 10128 zfcndac 10131 lfuhgr3 32664 bj-snsetex 34808 iooelexlt 35188 dford4 40463 relexp0eq 40895 |
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