Theorem cbvralsv 1967

Description: Change bound variable by using a substitution.

Assertion

Ref	Expression
cbvralsv	|- (A.x e. A ph <-> A.y e. A [y / x]ph)

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

Proof of Theorem cbvralsv

Step	Hyp	Ref	Expression
1		ax-17 971	. 2 |- (ph -> A.yph)
2		hbs1 1332	. 2 |- ([y / x]ph -> A.x[y / x]ph)
3		sbequ12 1181	. 2 |- (x = y -> (ph <-> [y / x]ph))
4	1, 2, 3	cbvral 1798	1 |- (A.x e. A ph <-> A.y e. A [y / x]ph)

Syntax hints:   <-> wb 146  [wsbc 1170  A.wral 1645

This theorem is referenced by:  ralxpf 3220

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-cleq 1469  df-clel 1472  df-ral 1649

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