Theorem csbid 1995

Description: Analog of sbid 1180 for proper substitution into a class.

Assertion

Ref	Expression
csbid	⊢ [x / x]A = A

Proof of Theorem csbid

Step	Hyp	Ref	Expression
1		df-csb 1992	. 2 ⊢ [x / x]A = {y∣[x / x]y ∈ A}
2		sbid 1180	. . 3 ⊢ ([x / x]y ∈ A ↔ y ∈ A)
3	2	abbii 1567	. 2 ⊢ {y∣[x / x]y ∈ A} = {y∣y ∈ A}
4		abid2 1572	. 2 ⊢ {y∣y ∈ A} = A
5	1, 3, 4	3eqtr 1491	1 ⊢ [x / x]A = A

Syntax hints:   = wceq 953   ∈ wcel 955  [wsbc 1166  {cab 1456  [csb 1991

This theorem is referenced by:  csbeq1a 1996

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-csb 1992

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