Theorem hbsbc1vOLD 2465

Description: Bound-variable hypothesis builder for class substitution.

Hypothesis

Ref	Expression
hbsbcv.1	|- A e. _V

Assertion

Ref	Expression
hbsbc1vOLD	|- ([A / x]ph -> A.x[A / x]ph)

Distinct variable group:   x,A

Proof of Theorem hbsbc1vOLD

Step	Hyp	Ref	Expression
1		ax-17 1317	. . 3 |- (y e. A -> A.x y e. A)
2	1	hbsbc1 2462	. 2 |- ((A e. _V -> [A / x]ph) -> A.x(A e. _V -> [A / x]ph))
3		hbsbcv.1	. . 3 |- A e. _V
4	3	a1bi 214	. 2 |- ([A / x]ph <-> (A e. _V -> [A / x]ph))
5	4	albii 1346	. 2 |- (A.x[A / x]ph <-> A.x(A e. _V -> [A / x]ph))
6	2, 4, 5	3imtr4i 236	1 |- ([A / x]ph -> A.x[A / x]ph)

Syntax hints:   -> wi 3  A.wal 1296   e. wcel 1300  [wsbc 1534  _Vcvv 2292

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

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