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Mirrors > Home > MPE Home > Th. List > 19.3v | Structured version Visualization version GIF version |
Description: Version of 19.3 2057 with a dv condition, requiring fewer axioms. Any formula can be universally quantified using a variable which it does not contain. See also 19.9v 1883. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 1922. (Revised by Wolf Lammen, 4-Dec-2017.) |
Ref | Expression |
---|---|
19.3v | ⊢ (∀𝑥𝜑 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alex 1743 | . 2 ⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑) | |
2 | 19.9v 1883 | . . 3 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ 𝜑) | |
3 | 2 | con2bii 346 | . 2 ⊢ (𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑) |
4 | 1, 3 | bitr4i 266 | 1 ⊢ (∀𝑥𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∀wal 1473 ∃wex 1695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 |
This theorem depends on definitions: df-bi 196 df-ex 1696 |
This theorem is referenced by: spvw 1885 19.27v 1895 19.28v 1896 19.37v 1897 axrep1 4700 kmlem14 8868 zfcndrep 9315 zfcndpow 9317 zfcndac 9320 bj-axrep1 31976 bj-snsetex 32144 iooelexlt 32386 dford4 36614 relexp0eq 37012 |
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