Theorem cbvrexsv 1971

Description: Change bound variable by using a substitution.

Assertion

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

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

Proof of Theorem cbvrexsv

Step	Hyp	Ref	Expression
1		ax-17 973	. 2 |- (ph -> A.yph)
2		hbs1 1334	. 2 |- ([y / x]ph -> A.x[y / x]ph)
3		sbequ12 1183	. 2 |- (x = y -> (ph <-> [y / x]ph))
4	1, 2, 3	cbvrex 1802	1 |- (E.x e. A ph <-> E.y e. A [y / x]ph)

Syntax hints:   <-> wb 146  [wsbc 1172  E.wrex 1649

This theorem is referenced by:  ac6sf 4770

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-cleq 1472  df-clel 1475  df-rex 1653

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