Theorem csbeq2d 2561

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 2457	. . . 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 1345	. . . 4 |- (ph -> A.x B = C)
5	1, 4	syl5 20	. . 3 |- (A e. D -> (ph -> [A / x]B = C))
6		sbceqdig 2554	. . 3 |- (A e. D -> ([A / x]B = C <-> [_A / x]_B = [_A / x]_C))
7	5, 6	sylibd 219	. 2 |- (A e. D -> (ph -> [_A / x]_B = [_A / x]_C))
8	7	impcom 378	1 |- ((ph /\ A e. D) -> [_A / x]_B = [_A / x]_C)

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  [wsbc 1534  [_csb 2540

This theorem is referenced by:  csbeq2dv 2562  csbnestg 2581  oprabval2gf 4955

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454  df-csb 2541

Copyright terms: Public domain