Theorem csbeq1 3139

Description: Analog of dfsbcq 3048 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Assertion

Ref	Expression
csbeq1	|- (A = B -> [_A / x]_C = [_B / x]_C)

Proof of Theorem csbeq1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq 3048	. . 3 |- (A = B -> ( [.A / x ].y e. C <-> [.B / x ].y e. C))
2	1	abbidv 2467	. 2 |- (A = B -> {y | [.A / x ].y e. C} = {y | [.B / x ].y e. C})
3		df-csb 3137	. 2 |- [_A / x]_C = {y | [.A / x ].y e. C}
4		df-csb 3137	. 2 |- [_B / x]_C = {y | [.B / x ].y e. C}
5	2, 3, 4	3eqtr4g 2410	1 |- (A = B -> [_A / x]_C = [_B / x]_C)

Syntax hints:   -> wi 4   = wceq 1642   e. wcel 1710  {cab 2339   [.wsbc 3046  [_csb 3136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-sbc 3047  df-csb 3137

This theorem is referenced by:  csbeq1d  3142  csbeq1a  3144  csbiebg  3175  sbcnestgf  3183  cbvralcsf  3198  cbvreucsf  3200  cbvrabcsf  3201  csbing  3462  csbifg  3690  csbiotag  4371  csbopabg  4637  sbcbrg  4685  csbima12g  4955  csbovg  5552  fvmpts  5701  fvmpt2i  5703  fvmptex  5721