Theorem sb56 1909

Description: Two equivalent ways of expressing the proper substitution of y for x in ph, when x and y are distinct. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 1786. (Contributed by NM, 14-Apr-2008.)

Assertion

Ref	Expression
sb56	|- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem sb56

Step	Hyp	Ref	Expression
1		hba1 1563	. 2 |- ( A. x ( x = y -> ph ) -> A. x A. x ( x = y -> ph ) )
2		ax11v 1850	. . 3 |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )
3		ax-4 1533	. . . 4 |- ( A. x ( x = y -> ph ) -> ( x = y -> ph ) )
4	3	com12 30	. . 3 |- ( x = y -> ( A. x ( x = y -> ph ) -> ph ) )
5	2, 4	impbid 129	. 2 |- ( x = y -> ( ph <-> A. x ( x = y -> ph ) ) )
6	1, 5	equsex 1751	1 |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1371   E.wex 1515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  sb6  1910  sb5  1911  alexeq  2899  dfdif3  3283