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Theorem ax16i 2046
 Description: Inference with ax16 2045 as its conclusion. (Contributed by NM, 20-May-2008.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
ax16i.1 (x = z → (φψ))
ax16i.2 (ψxψ)
Assertion
Ref Expression
ax16i (x x = y → (φxφ))
Distinct variable groups:   x,y,z   φ,z
Allowed substitution hints:   φ(x,y)   ψ(x,y,z)

Proof of Theorem ax16i
StepHypRef Expression
1 nfv 1619 . . 3 z x = y
2 nfv 1619 . . 3 x z = y
3 ax-8 1675 . . 3 (x = z → (x = yz = y))
41, 2, 3cbv3 1982 . 2 (x x = yz z = y)
5 ax-8 1675 . . . . 5 (z = x → (z = yx = y))
65spimv 1990 . . . 4 (z z = yx = y)
7 equcomi 1679 . . . . . 6 (x = yy = x)
8 equcomi 1679 . . . . . . 7 (z = yy = z)
9 ax-8 1675 . . . . . . 7 (y = z → (y = xz = x))
108, 9syl 15 . . . . . 6 (z = y → (y = xz = x))
117, 10syl5com 26 . . . . 5 (x = y → (z = yz = x))
1211alimdv 1621 . . . 4 (x = y → (z z = yz z = x))
136, 12mpcom 32 . . 3 (z z = yz z = x)
14 equcomi 1679 . . . 4 (z = xx = z)
1514alimi 1559 . . 3 (z z = xz x = z)
1613, 15syl 15 . 2 (z z = yz x = z)
17 ax16i.1 . . . . 5 (x = z → (φψ))
1817biimpcd 215 . . . 4 (φ → (x = zψ))
1918alimdv 1621 . . 3 (φ → (z x = zzψ))
20 ax16i.2 . . . . 5 (ψxψ)
2120nfi 1551 . . . 4 xψ
22 nfv 1619 . . . 4 zφ
2317biimprd 214 . . . . 5 (x = z → (ψφ))
2414, 23syl 15 . . . 4 (z = x → (ψφ))
2521, 22, 24cbv3 1982 . . 3 (zψxφ)
2619, 25syl6com 31 . 2 (z x = z → (φxφ))
274, 16, 263syl 18 1 (x x = y → (φxφ))
 Colors of variables: wff setvar class Syntax hints:   → 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:  ax16ALT  2047
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