Theorem sb56 1272

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 1178.

Assertion

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

Distinct variable group:   x,y

Proof of Theorem sb56

Step	Hyp	Ref	Expression
1		hba1 1009	. 2 |- (A.x(x = y -> ph) -> A.xA.x(x = y -> ph))
2		ax11v 1271	. . 3 |- (x = y -> (ph -> A.x(x = y -> ph)))
3		ax-4 977	. . . 4 |- (A.x(x = y -> ph) -> (x = y -> ph))
4	3	com12 11	. . 3 |- (x = y -> (A.x(x = y -> ph) -> ph))
5	2, 4	impbid 519	. 2 |- (x = y -> (ph <-> A.x(x = y -> ph)))
6	1, 5	equsex 1158	1 |- (E.x(x = y /\ ph) <-> A.x(x = y -> ph))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 958   = wceq 960  E.wex 984

This theorem is referenced by:  sb6 1273  sb5 1274  alexeq 1892

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 967  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-16 1216  ax-11o 1224

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 985

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