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Theorem ax10lem2 1937
 Description: Lemma for ax10 1944. Change free variable. (Contributed by NM, 25-Jul-2015.)
Assertion
Ref Expression
ax10lem2 (x x = yx x = z)
Distinct variable groups:   x,y   x,z

Proof of Theorem ax10lem2
StepHypRef Expression
1 hbe1 1731 . . . 4 (x ¬ x = yxx ¬ x = y)
2 equequ2 1686 . . . . . . . 8 (z = y → (x = zx = y))
32biimprd 214 . . . . . . 7 (z = y → (x = yx = z))
43con3rr3 128 . . . . . 6 x = z → (z = y → ¬ x = y))
5 19.8a 1756 . . . . . 6 x = yx ¬ x = y)
64, 5syl6 29 . . . . 5 x = z → (z = yx ¬ x = y))
7 ax-17 1616 . . . . . 6 z = yx ¬ z = y)
8 equequ1 1684 . . . . . . . 8 (x = z → (x = yz = y))
98notbid 285 . . . . . . 7 (x = z → (¬ x = y ↔ ¬ z = y))
109biimprd 214 . . . . . 6 (x = z → (¬ z = y → ¬ x = y))
117, 10spimeh 1667 . . . . 5 z = yx ¬ x = y)
126, 11pm2.61d1 151 . . . 4 x = zx ¬ x = y)
131, 12exlimih 1804 . . 3 (x ¬ x = zx ¬ x = y)
14 exnal 1574 . . 3 (x ¬ x = z ↔ ¬ x x = z)
15 exnal 1574 . . 3 (x ¬ x = y ↔ ¬ x x = y)
1613, 14, 153imtr3i 256 . 2 x x = z → ¬ x x = y)
1716con4i 122 1 (x x = yx x = z)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545 This theorem is referenced by:  ax10lem3  1938
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