Theorem csbeq2d 2008

Description: Formula-building deduction rule for class substitution.

Hypotheses

Ref	Expression
csbeq2d.1	|- (ph -> A.xph)
csbeq2d.2	|- (ph -> B = C)

Assertion

Ref	Expression
csbeq2d	|- ((ph /\ A e. D) -> [_A / x]_B = [_A / x]_C)

Proof of Theorem csbeq2d

Step	Hyp	Ref	Expression
1		a4sbc 1935	. . . 4 |- (A e. D -> (A.x B = C -> [A / x]B = C))
2		csbeq2d.1	. . . . 5 |- (ph -> A.xph)
3		csbeq2d.2	. . . . 5 |- (ph -> B = C)
4	2, 3	19.21ai 995	. . . 4 |- (ph -> A.x B = C)
5	1, 4	syl5 21	. . 3 |- (A e. D -> (ph -> [A / x]B = C))
6		sbceqdig 2002	. . 3 |- (A e. D -> ([A / x]B = C <-> [_A / x]_B = [_A / x]_C))
7	5, 6	sylibd 202	. 2 |- (A e. D -> (ph -> [_A / x]_B = [_A / x]_C))
8	7	impcom 351	1 |- ((ph /\ A e. D) -> [_A / x]_B = [_A / x]_C)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  [wsbc 1166  [_csb 1991

This theorem is referenced by:  csbeq2dv 2009  csbnestg 2026  oprabval2gf 4011

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-9 962  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-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-sbc 1932  df-csb 1992

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