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Theorem ax9 1949
 Description: Theorem showing that ax-9 1654 follows from the weaker version ax9v 1655. (Even though this theorem depends on ax-9 1654, all references of ax-9 1654 are made via ax9v 1655. An earlier version stated ax9v 1655 as a separate axiom, but having two axioms caused some confusion.) This theorem should be referenced in place of ax-9 1654 so that all proofs can be traced back to ax9v 1655. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.)
Assertion
Ref Expression
ax9 ¬ x ¬ x = y

Proof of Theorem ax9
Dummy variable v is distinct from all other variables.
StepHypRef Expression
1 sp 1747 . . 3 (x ¬ x = y → ¬ x = y)
2 sp 1747 . . 3 (x x = yx = y)
31, 2nsyl3 111 . 2 (x x = y → ¬ x ¬ x = y)
4 ax9v 1655 . . 3 ¬ v ¬ v = y
5 dveeq2 1940 . . . . . 6 x x = y → (v = yx v = y))
6 ax9v 1655 . . . . . . 7 ¬ x ¬ x = v
7 hba1 1786 . . . . . . . 8 (x v = yxx v = y)
8 sp 1747 . . . . . . . . . 10 (x v = yv = y)
9 equequ2 1686 . . . . . . . . . 10 (v = y → (x = vx = y))
108, 9syl 15 . . . . . . . . 9 (x v = y → (x = vx = y))
1110notbid 285 . . . . . . . 8 (x v = y → (¬ x = v ↔ ¬ x = y))
127, 11albidh 1590 . . . . . . 7 (x v = y → (x ¬ x = vx ¬ x = y))
136, 12mtbii 293 . . . . . 6 (x v = y → ¬ x ¬ x = y)
145, 13syl6com 31 . . . . 5 (v = y → (¬ x x = y → ¬ x ¬ x = y))
1514con3i 127 . . . 4 (¬ (¬ x x = y → ¬ x ¬ x = y) → ¬ v = y)
1615alrimiv 1631 . . 3 (¬ (¬ x x = y → ¬ x ¬ x = y) → v ¬ v = y)
174, 16mt3 171 . 2 x x = y → ¬ x ¬ x = y)
183, 17pm2.61i 156 1 ¬ x ¬ x = y
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545 This theorem is referenced by:  ax9o  1950  a9e  1951
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