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Theorem sb8h 1731
 Description: Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
Hypothesis
Ref Expression
sb8h.1 (φyφ)
Assertion
Ref Expression
sb8h (xφy[y / x]φ)

Proof of Theorem sb8h
StepHypRef Expression
1 sb8h.1 . 2 (φyφ)
21hbsb3 1686 . 2 ([y / x]φx[y / x]φ)
3 sbequ12 1651 . 2 (x = y → (φ ↔ [y / x]φ))
41, 2, 3cbvalh 1633 1 (xφy[y / x]φ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  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:  sbhb  1813  sb8euh  1920
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