Theorem sbcieg 2484

Description: Conversion of implicit substitution to explicit class substitution.

Hypothesis

Ref	Expression
sbcieg.1	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
sbcieg	|- (A e. B -> ([A / x]ph <-> ps))

Distinct variable groups:   x,A   ps,x

Proof of Theorem sbcieg

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (A e. B -> A e. _V)
2		ax-17 1317	. . . 4 |- (ps -> A.xps)
3	2	a1i 8	. . 3 |- (A e. _V -> (ps -> A.xps))
4		sbcieg.1	. . 3 |- (x = A -> (ph <-> ps))
5	3, 4	sbciegf 2483	. 2 |- (A e. _V -> ([A / x]ph <-> ps))
6	1, 5	syl 12	1 |- (A e. B -> ([A / x]ph <-> ps))

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298   e. wcel 1300  [wsbc 1534  _Vcvv 2292

This theorem is referenced by:  sbcie 2485  sbceqal 2502  trsbc 5843  bnj527OLD 12526  bnj1462 13546  trsbcVD 16701  joinval2 16816  meetval2 16823

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-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454

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