Theorem sb7 1322

Description: An alternate definition of proper substitution df-sb 1155. By introducing a dummy variable z in the definiens, we are able to eliminate any distinct variable restrictions among the variables x, y, and ph of the definiendum. No distinct variable conflicts arise because z effectively insulates x from y. To achieve this, we use a chain of two substitutions in the form of sb5 1252, first z for x then y for z. Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 1441. Theorem sb7f 1323 provides a version where ph and z don't have to be distinct.

Assertion

Ref	Expression
sb7	|- ([y / x]ph <-> E.z(z = y /\ E.x(x = z /\ ph)))

Distinct variable groups:   x,z   y,z   ph,z

Proof of Theorem sb7

Step	Hyp	Ref	Expression
1		sb5 1252	. . 3 |- ([z / x]ph <-> E.x(x = z /\ ph))
2	1	sbbii 1157	. 2 |- ([y / z][z / x]ph <-> [y / z]E.x(x = z /\ ph))
3		ax-17 1190	. . 3 |- (ph -> A.zph)
4	3	sbco2 1239	. 2 |- ([y / z][z / x]ph <-> [y / x]ph)
5		sb5 1252	. 2 |- ([y / z]E.x(x = z /\ ph) <-> E.z(z = y /\ E.x(x = z /\ ph)))
6	2, 4, 5	3bitr3 181	1 |- ([y / x]ph <-> E.z(z = y /\ E.x(x = z /\ ph)))

Syntax hints:   <-> wb 146   /\ wa 223  E.wex 956   = wceq 1099  [wsbc 1153

This theorem is referenced by:  sb7f 1323

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155

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