Theorem csbeq2gOLD 38765

Description: Formula-building implication rule for class substitution. Closed form of csbeq2i 3993. csbeq2gOLD 38765 is derived from the virtual deduction proof csbeq2gVD 39128. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete version of csbeq2 3537 as of 11-Oct-2018. (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
csbeq2gOLD	|- ( A e. V -> ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C ) )

Proof of Theorem csbeq2gOLD

Step	Hyp	Ref	Expression
1		spsbc 3448	. 2 |- ( A e. V -> ( A. x B = C -> [. A / x ]. B = C ) )
2		sbceqg 3984	. 2 |- ( A e. V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) )
3	1, 2	sylibd 229	1 |- ( A e. V -> ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C ) )

Syntax hints:   -> wi 4   A.wal 1481   = wceq 1483   e. wcel 1990   [.wsbc 3435   [_csb 3533

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-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534

This theorem is referenced by:  csbsngVD  39129  csbxpgVD  39130  csbresgVD  39131  csbrngVD  39132  csbima12gALTVD  39133