Theorem cbvexsv 38762

Description: A theorem pertaining to the substitution for an existentially quantified variable when the substituted variable does not occur in the quantified wff. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   ph, y

Allowed substitution hint:   ph( x)

Proof of Theorem cbvexsv

Step	Hyp	Ref	Expression
1		cbvrexsv 3183	. 2 |- ( E. x e. _V ph <-> E. y e. _V [ y / x ] ph )
2		rexv 3220	. 2 |- ( E. x e. _V ph <-> E. x ph )
3		rexv 3220	. 2 |- ( E. y e. _V [ y / x ] ph <-> E. y [ y / x ] ph )
4	1, 2, 3	3bitr3i 290	1 |- ( E. x ph <-> E. y [ y / x ] ph )

Syntax hints:   <-> wb 196   E.wex 1704   [wsb 1880   E.wrex 2913   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202

This theorem is referenced by:  onfrALTlem1  38763  onfrALTlem1VD  39126