Theorem csb2 3138

Description: Alternate expression for the proper substitution into a class, without referencing substitution into a wff. Note that x can be free in B but cannot occur in A. (Contributed by NM, 2-Dec-2013.)

Assertion

Ref	Expression
csb2	⊢ [A / x]B = {y ∣ ∃x(x = A ∧ y ∈ B)}

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

Allowed substitution hint:   B(x)

Proof of Theorem csb2

Step	Hyp	Ref	Expression
1		df-csb 3137	. 2 ⊢ [A / x]B = {y ∣ [̣A / x]̣y ∈ B}
2		sbc5 3070	. . 3 ⊢ ([̣A / x]̣y ∈ B ↔ ∃x(x = A ∧ y ∈ B))
3	2	abbii 2465	. 2 ⊢ {y ∣ [̣A / x]̣y ∈ B} = {y ∣ ∃x(x = A ∧ y ∈ B)}
4	1, 3	eqtri 2373	1 ⊢ [A / x]B = {y ∣ ∃x(x = A ∧ y ∈ B)}

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  [̣wsbc 3046  [csb 3136

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-or 359  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-nfc 2478  df-v 2861  df-sbc 3047  df-csb 3137

This theorem is referenced by: (None)