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Mirrors > Home > MPE Home > Th. List > ax6 | Structured version Visualization version GIF version |
Description: Theorem showing that ax-6 1945
follows from the weaker version ax6v 1946.
(Even though this theorem depends on ax-6 1945,
all references of ax-6 1945 are
made via ax6v 1946. An earlier version stated ax6v 1946
as a separate axiom,
but having two axioms caused some confusion.)
This theorem should be referenced in place of ax-6 1945 so that all proofs can be traced back to ax6v 1946. When possible, use the weaker ax6v 1946 rather than ax6 2287 since the ax6v 1946 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 2286 | . 2 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | df-ex 1745 | . 2 ⊢ (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦) | |
3 | 1, 2 | mpbi 220 | 1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∀wal 1521 ∃wex 1744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-12 2087 ax-13 2282 |
This theorem depends on definitions: df-bi 197 df-an 385 df-ex 1745 |
This theorem is referenced by: axc10 2288 |
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