Theorem sbceqi 33913

Description: Distribution of class substitution over equality, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)

Hypotheses

Ref	Expression
sbceqi.1	|- A e. _V
sbceqi.2	|- [_ A / x ]_ B = D
sbceqi.3	|- [_ A / x ]_ C = E

Assertion

Ref	Expression
sbceqi	|- ( [. A / x ]. B = C <-> D = E )

Proof of Theorem sbceqi

Step	Hyp	Ref	Expression
1		sbceqi.1	. . 3 |- A e. _V
2		sbceqg 3984	. . 3 |- ( A e. _V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) )
3	1, 2	ax-mp 5	. 2 |- ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C )
4		sbceqi.2	. . 3 |- [_ A / x ]_ B = D
5		sbceqi.3	. . 3 |- [_ A / x ]_ C = E
6	4, 5	eqeq12i 2636	. 2 |- ( [_ A / x ]_ B = [_ A / x ]_ C <-> D = E )
7	3, 6	bitri 264	1 |- ( [. A / x ]. B = C <-> D = E )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   _Vcvv 3200   [.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:  sbccom2lem  33929