Theorem hbsbc1v 1940

Description: Bound-variable hypothesis builder for class substitution.

Hypothesis

Ref	Expression
hbsbcv.1	|- A e. V

Assertion

Ref	Expression
hbsbc1v	|- ([A / x]ph -> A.x[A / x]ph)

Distinct variable group:   x,A

Proof of Theorem hbsbc1v

Step	Hyp	Ref	Expression
1		ax-17 968	. . 3 |- (y e. A -> A.x y e. A)
2	1	hbsbc1 1939	. 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 197	. 2 |- ([A / x]ph <-> (A e. V -> [A / x]ph))
5	4	albii 996	. 2 |- (A.x[A / x]ph <-> A.x(A e. V -> [A / x]ph))
6	2, 4, 5	3imtr4 219	1 |- ([A / x]ph -> A.x[A / x]ph)

Syntax hints:   -> wi 3  A.wal 951   e. wcel 955  [wsbc 1166  Vcvv 1802

This theorem is referenced by:  findes 3150  tfindes 3154  dfopab2 4097  dfoprab3 4098  nn1suc 5887  uzindOLD 6156  nn0ind-raph 6162  uzind4s 6384  fzrevralt 6451  fsum1f 6945  fsump1f 6949

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-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-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-sbc 1932

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