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Mirrors > Home > MPE Home > Th. List > ax6 | Structured version Visualization version GIF version |
Description: Theorem showing that ax-6 2077
follows from the weaker version ax6v 2078.
(Even though this theorem depends on ax-6 2077,
all references of ax-6 2077 are
made via ax6v 2078. An earlier version stated ax6v 2078
as a separate axiom,
but having two axioms caused some confusion.)
This theorem should be referenced in place of ax-6 2077 so that all proofs can be traced back to ax6v 2078. When possible, use the weaker ax6v 2078 rather than ax6 2404 since the ax6v 2078 derivation is much shorter and requires fewer axioms. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 4-Feb-2018.) |
Ref | Expression |
---|---|
ax6 | ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax6e 2403 | . 2 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | df-ex 1881 | . 2 ⊢ (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦) | |
3 | 1, 2 | mpbi 222 | 1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∀wal 1656 ∃wex 1880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-12 2222 ax-13 2390 |
This theorem depends on definitions: df-bi 199 df-an 387 df-ex 1881 |
This theorem is referenced by: axc10 2405 |
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