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Theorem ax16ALT2 2048
 Description: Alternate proof of ax16 2045. (Contributed by NM, 8-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax16ALT2 (x x = y → (φxφ))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem ax16ALT2
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 aev 1991 . 2 (x x = yz x = z)
2 sbequ12 1919 . . . . 5 (x = z → (φ ↔ [z / x]φ))
32biimpcd 215 . . . 4 (φ → (x = z → [z / x]φ))
43alimdv 1621 . . 3 (φ → (z x = zz[z / x]φ))
5 nfv 1619 . . . . 5 zφ
65nfs1 2044 . . . 4 x[z / x]φ
7 stdpc7 1917 . . . 4 (z = x → ([z / x]φφ))
86, 5, 7cbv3 1982 . . 3 (z[z / x]φxφ)
94, 8syl6com 31 . 2 (z x = z → (φxφ))
101, 9syl 15 1 (x x = y → (φxφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540  [wsb 1648 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-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  df-sb 1649 This theorem is referenced by:  a16gALT  2049
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