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Theorem ax16 1691
Description: Theorem showing that ax-16 1692 is redundant if ax-17 1416 is included in the axiom system. The important part of the proof is provided by aev 1690.

See ax16ALT 1736 for an alternate proof that does not require ax-10 1393 or ax-12 1399.

This theorem should not be referenced in any proof. Instead, use ax-16 1692 below so that theorems needing ax-16 1692 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 1690 . 2 (x x = yz x = z)
2 ax-17 1416 . . . 4 (φzφ)
3 sbequ12 1651 . . . . 5 (x = z → (φ ↔ [z / x]φ))
43biimpcd 148 . . . 4 (φ → (x = z → [z / x]φ))
52, 4alimdh 1353 . . 3 (φ → (z x = zz[z / x]φ))
62hbsb3 1686 . . . 4 ([z / x]φx[z / x]φ)
7 stdpc7 1650 . . . 4 (z = x → ([z / x]φφ))
86, 2, 7cbv3h 1628 . . 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 1240  [wsb 1642
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643
This theorem is referenced by:  dveeq2  1693  dveeq2or  1694  a16g  1741  exists2  1994
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