Theorem r19.21ad 1261

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.)

Hypotheses

Ref	Expression
r19.21ad.1	|- (ph -> A.xph)
r19.21ad.2	|- (ps -> A.xps)
r19.21ad.3	|- (ph -> (ps -> (x e. A -> ch)))

Assertion

Ref	Expression
r19.21ad	|- (ph -> (ps -> A.x e. A ch))

Proof of Theorem r19.21ad

Step	Hyp	Ref	Expression
1		r19.21ad.1	. . 3 |- (ph -> A.xph)
2		r19.21ad.2	. . 3 |- (ps -> A.xps)
3		r19.21ad.3	. . 3 |- (ph -> (ps -> (x e. A -> ch)))
4	1, 2, 3	19.21ad 741	. 2 |- (ph -> (ps -> A.x(x e. A -> ch)))
5		df-ral 1205	. 2 |- (A.x e. A ch <-> A.x(x e. A -> ch))
6	4, 5	syl6ibr 186	1 |- (ph -> (ps -> A.x e. A ch))

Syntax hints:   -> wi 2  A.wal 672   e. wcel 1092  A.wral 1201

This theorem is referenced by:  r19.21adv 1262  isotrALT 2936  tfrlem1 2949  mapxpen 3390

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ral 1205

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