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Theorem ax11inda2ALT 2198
 Description: A proof of ax11inda2 2199 that is slightly more direct. (Contributed by NM, 4-May-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ax11inda2.1 x x = y → (x = y → (φx(x = yφ))))
Assertion
Ref Expression
ax11inda2ALT x x = y → (x = y → (zφx(x = yzφ))))
Distinct variable group:   y,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem ax11inda2ALT
StepHypRef Expression
1 ax-1 5 . . . . . . . 8 (xφ → (x = yxφ))
21a5i-o 2150 . . . . . . 7 (xφx(x = yxφ))
32a1i 10 . . . . . 6 (z z = x → (xφx(x = yxφ)))
4 biidd 228 . . . . . . 7 (z z = x → (φφ))
54dral1-o 2154 . . . . . 6 (z z = x → (zφxφ))
65imbi2d 307 . . . . . . 7 (z z = x → ((x = yzφ) ↔ (x = yxφ)))
76dral2-o 2181 . . . . . 6 (z z = x → (x(x = yzφ) ↔ x(x = yxφ)))
83, 5, 73imtr4d 259 . . . . 5 (z z = x → (zφx(x = yzφ)))
98aecoms-o 2152 . . . 4 (x x = z → (zφx(x = yzφ)))
109a1d 22 . . 3 (x x = z → (x = y → (zφx(x = yzφ))))
1110a1d 22 . 2 (x x = z → (¬ x x = y → (x = y → (zφx(x = yzφ)))))
12 simplr 731 . . . . 5 (((¬ x x = z ¬ x x = y) x = y) → ¬ x x = y)
13 dveeq1-o 2187 . . . . . . . 8 z z = x → (x = yz x = y))
1413naecoms-o 2178 . . . . . . 7 x x = z → (x = yz x = y))
1514imp 418 . . . . . 6 ((¬ x x = z x = y) → z x = y)
1615adantlr 695 . . . . 5 (((¬ x x = z ¬ x x = y) x = y) → z x = y)
17 hbnae-o 2179 . . . . . . 7 x x = yz ¬ x x = y)
18 hba1-o 2149 . . . . . . 7 (z x = yzz x = y)
1917, 18hban 1828 . . . . . 6 ((¬ x x = y z x = y) → zx x = y z x = y))
20 ax-4 2135 . . . . . . 7 (z x = yx = y)
21 ax11inda2.1 . . . . . . . 8 x x = y → (x = y → (φx(x = yφ))))
2221imp 418 . . . . . . 7 ((¬ x x = y x = y) → (φx(x = yφ)))
2320, 22sylan2 460 . . . . . 6 ((¬ x x = y z x = y) → (φx(x = yφ)))
2419, 23alimdh 1563 . . . . 5 ((¬ x x = y z x = y) → (zφzx(x = yφ)))
2512, 16, 24syl2anc 642 . . . 4 (((¬ x x = z ¬ x x = y) x = y) → (zφzx(x = yφ)))
26 ax-7 1734 . . . . . 6 (zx(x = yφ) → xz(x = yφ))
27 hbnae-o 2179 . . . . . . 7 x x = zx ¬ x x = z)
28 hbnae-o 2179 . . . . . . . . 9 x x = zz ¬ x x = z)
2928, 14nfdh 1767 . . . . . . . 8 x x = z → Ⅎz x = y)
30 19.21t 1795 . . . . . . . 8 (Ⅎz x = y → (z(x = yφ) ↔ (x = yzφ)))
3129, 30syl 15 . . . . . . 7 x x = z → (z(x = yφ) ↔ (x = yzφ)))
3227, 31albidh 1590 . . . . . 6 x x = z → (xz(x = yφ) ↔ x(x = yzφ)))
3326, 32syl5ib 210 . . . . 5 x x = z → (zx(x = yφ) → x(x = yzφ)))
3433ad2antrr 706 . . . 4 (((¬ x x = z ¬ x x = y) x = y) → (zx(x = yφ) → x(x = yzφ)))
3525, 34syld 40 . . 3 (((¬ x x = z ¬ x x = y) x = y) → (zφx(x = yzφ)))
3635exp31 587 . 2 x x = z → (¬ x x = y → (x = y → (zφx(x = yzφ)))))
3711, 36pm2.61i 156 1 x x = y → (x = y → (zφx(x = yzφ))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544 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  ax-4 2135  ax-5o 2136  ax-6o 2137  ax-10o 2139  ax-12o 2142 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: (None)
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