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Theorem sb56 2172
Description: Two equivalent ways of expressing the proper substitution of for in , when and are distinct. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 1740. The implication "to the left" is equs4 2035 and does not require any dv condition. Theorem equs45f 2091 replaces the dv condition with a non-freeness hypothesis. (Contributed by NM, 14-Apr-2008.)
Assertion
Ref Expression
sb56
Distinct variable group:   ,

Proof of Theorem sb56
StepHypRef Expression
1 nfa1 1897 . 2
2 ax12v 1855 . . 3
3 sp 1859 . . . 4
43com12 31 . . 3
52, 4impbid 191 . 2
61, 5equsex 2038 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612
This theorem is referenced by:  sb6  2173  sb5  2174  mopick  2356  alexeqg  3228  alexeq  3229  pm13.196a  31321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-12 1854  ax-13 1999
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-nf 1617
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