Theorem csbeq1 3140

Description: Analog of dfsbcq 3049 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 3049	. . 3 ⊢ (A = B → ([̣A / x]̣y ∈ C ↔ [̣B / x]̣y ∈ C))
2	1	abbidv 2468	. 2 ⊢ (A = B → {y ∣ [̣A / x]̣y ∈ C} = {y ∣ [̣B / x]̣y ∈ C})
3		df-csb 3138	. 2 ⊢ [A / x]C = {y ∣ [̣A / x]̣y ∈ C}
4		df-csb 3138	. 2 ⊢ [B / x]C = {y ∣ [̣B / x]̣y ∈ C}
5	2, 3, 4	3eqtr4g 2410	1 ⊢ (A = B → [A / x]C = [B / x]C)

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {cab 2339  [̣wsbc 3047  [csb 3137

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 3048  df-csb 3138

This theorem is referenced by:  csbeq1d  3143  csbeq1a  3145  csbiebg  3176  sbcnestgf  3184  cbvralcsf  3199  cbvreucsf  3201  cbvrabcsf  3202  csbing  3463  csbifg  3691  csbiotag  4372  csbopabg  4638  sbcbrg  4686  csbima12g  4956  csbovg  5553  fvmpts  5702  fvmpt2i  5704  fvmptex  5722