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Theorem ax16 1676
Description: Theorem showing that ax-16 1677 is redundant if ax-17 1400 is included in the axiom system. The important part of the proof is provided by aev 1675.

See ax16ALT 1721 for an alternate proof that does not require ax-10 1377 or ax-12 1383.

This theorem should not be referenced in any proof. Instead, use ax-16 1677 below so that theorems needing ax-16 1677 can be more easily identified. (Contributed by NM, 8-Nov-2006.)

Assertion
Ref Expression
ax16 (x x = y → (φxφ))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem ax16
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 aev 1675 . 2 (x x = yz x = z)
2 ax-17 1400 . . . 4 (φzφ)
3 sbequ12 1636 . . . . 5 (x = z → (φ ↔ [z / x]φ))
43biimpcd 148 . . . 4 (φ → (x = z → [z / x]φ))
52, 4alimdh 1336 . . 3 (φ → (z x = zz[z / x]φ))
62hbsb3 1671 . . . 4 ([z / x]φx[z / x]φ)
7 stdpc7 1635 . . . 4 (z = x → ([z / x]φφ))
86, 2, 7cbv3h 1613 . . 3 (z[z / x]φxφ)
95, 8syl6com 31 . 2 (z x = z → (φxφ))
101, 9syl 14 1 (x x = y → (φxφ))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1226  [wsb 1627
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628
This theorem is referenced by:  dveeq2  1678  dveeq2or  1679  a16g  1726  exists2  1979
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