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Theorem ax11inda2 2199
 Description: Induction step for constructing a substitution instance of ax-11o 2141 without using ax-11o 2141. Quantification case. When z and y are distinct, this theorem avoids the dummy variables needed by the more general ax11inda 2200. (Contributed by NM, 24-Jan-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
ax11inda2 x x = y → (x = y → (zφx(x = yzφ))))
Distinct variable group:   y,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem ax11inda2
StepHypRef Expression
1 ax-1 5 . . . . 5 (zφ → (x = yzφ))
2 a16g-o 2186 . . . . 5 (y y = z → ((x = yzφ) → x(x = yzφ)))
31, 2syl5 28 . . . 4 (y y = z → (zφx(x = yzφ)))
43a1d 22 . . 3 (y y = z → (x = y → (zφx(x = yzφ))))
54a1d 22 . 2 (y y = z → (¬ x x = y → (x = y → (zφx(x = yzφ)))))
6 ax11inda2.1 . . 3 x x = y → (x = y → (φx(x = yφ))))
76ax11indalem 2197 . 2 y y = z → (¬ x x = y → (x = y → (zφx(x = yzφ)))))
85, 7pm2.61i 156 1 x x = y → (x = y → (zφx(x = yzφ))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀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  ax-4 2135  ax-5o 2136  ax-6o 2137  ax-10o 2139  ax-12o 2142  ax-16 2144 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:  ax11inda  2200
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