Theorem sbab 1188

Description: The right-hand side of the second equality is a way of representing proper substitution of y for x into a class variable.

Assertion

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

Distinct variable group(s):   z,A   x,z   y,z

Proof of Theorem sbab

Step	Hyp	Ref	Expression
1		sbequ12 865	. 2 ⊢ (x = y → (z ∈ A ↔ [y / x]z ∈ A))
2	1	biabrdv 1184	1 ⊢ (x = y → A = {z∣[y / x]z ∈ A})

Syntax hints:   → wi 2   = weq 797  [wsb 852  {cab 1090   = wceq 1091   ∈ wcel 1092

This theorem is referenced by:  moop2 1910  fvopabgf 2874  fvopabnf 2875  oprabval4g 3053  seqlem1 4662

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099

metamath.org