Theorem ceqsal 1817

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis.

Hypotheses

Ref	Expression
ceqsal.1	|- (ps -> A.xps)
ceqsal.2	|- A e. V
ceqsal.3	|- (x = A -> (ph <-> ps))

Assertion

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

Distinct variable group:   x,A

Proof of Theorem ceqsal

Step	Hyp	Ref	Expression
1		ceqsal.2	. 2 |- A e. V
2		ceqsal.1	. . 3 |- (ps -> A.xps)
3		ceqsal.3	. . 3 |- (x = A -> (ph <-> ps))
4	2, 3	ceqsalg 1816	. 2 |- (A e. V -> (A.x(x = A -> ph) <-> ps))
5	1, 4	ax-mp 7	1 |- (A.x(x = A -> ph) <-> ps)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 951   = wceq 953   e. wcel 955  Vcvv 1802

This theorem is referenced by:  ceqsalv 1818  csbieb 2020

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  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

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