Theorem eqsbc3 3475

Description: Substitution applied to an atomic wff. Set theory version of eqsb3 2728. (Contributed by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
eqsbc3	|- ( A e. V -> ( [. A / x ]. x = B <-> A = B ) )

Distinct variable group:   x, B

Allowed substitution hints:   A( x)   V( x)

Proof of Theorem eqsbc3

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq 3437	. 2 |- ( y = A -> ( [. y / x ]. x = B <-> [. A / x ]. x = B ) )
2		eqeq1 2626	. 2 |- ( y = A -> ( y = B <-> A = B ) )
3		sbsbc 3439	. . 3 |- ( [ y / x ] x = B <-> [. y / x ]. x = B )
4		eqsb3 2728	. . 3 |- ( [ y / x ] x = B <-> y = B )
5	3, 4	bitr3i 266	. 2 |- ( [. y / x ]. x = B <-> y = B )
6	1, 2, 5	vtoclbg 3267	1 |- ( A e. V -> ( [. A / x ]. x = B <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   [wsb 1880   e. wcel 1990   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by:  sbceqal  3487  eqsbc3r  3492  eqsbc3rOLD  3493  fmptsnd  6435  fvmptnn04if  20654  snfil  21668  f1omptsnlem  33183  mptsnunlem  33185  topdifinffinlem  33195  relowlpssretop  33212  iotavalb  38631  onfrALTlem5  38757  eqsbc3rVD  39075  onfrALTlem5VD  39121