Theorem sbab 2475

Description: The right-hand side of the second equality is a way of representing proper substitution of y for x into a class variable. (Contributed by NM, 14-Sep-2003.)

Assertion

Ref	Expression
sbab	⊢ (x = y → A = {z ∣ [y / x]z ∈ A})

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

Allowed substitution hints:   A(x,y)

Proof of Theorem sbab

Step	Hyp	Ref	Expression
1		sbequ12 1919	. 2 ⊢ (x = y → (z ∈ A ↔ [y / x]z ∈ A))
2	1	abbi2dv 2468	1 ⊢ (x = y → A = {z ∣ [y / x]z ∈ A})

Syntax hints:   → wi 4   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339

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

This theorem is referenced by:  sbcel12g  3151  sbceqg  3152