Theorem sbceq2dig 2006

Description: Move proper substitution to second argument of an equality.

Assertion

Ref	Expression
sbceq2dig	|- (A e. D -> ([A / x]B = C <-> B = [_A / x]_C))

Distinct variable group:   x,B

Proof of Theorem sbceq2dig

Step	Hyp	Ref	Expression
1		sbceqdig 2002	. 2 |- (A e. D -> ([A / x]B = C <-> [_A / x]_B = [_A / x]_C))
2		ax-17 968	. . . 4 |- (y e. B -> A.x y e. B)
3	2	csbconstgf 2000	. . 3 |- (A e. D -> [_A / x]_B = B)
4	3	eqeq1d 1475	. 2 |- (A e. D -> ([_A / x]_B = [_A / x]_C <-> B = [_A / x]_C))
5	1, 4	bitrd 526	1 |- (A e. D -> ([A / x]B = C <-> B = [_A / x]_C))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 953   e. wcel 955  [wsbc 1166  [_csb 1991

This theorem is referenced by:  dfoprab5 4099  foprab2 4103  eqerlem 4254  fsumcnlem 7923

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-sbc 1932  df-csb 1992

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