Theorem r19.21ad 2180

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 1406	. 2 |- (ph -> (ps -> A.x(x e. A -> ch)))
5		df-ral 2109	. 2 |- (A.x e. A ch <-> A.x(x e. A -> ch))
6	4, 5	syl6ibr 230	1 |- (ph -> (ps -> A.x e. A ch))

Syntax hints:   -> wi 3  A.wal 1296   e. wcel 1300  A.wral 2105

This theorem is referenced by:  r19.21adv 2181  ssralv2 2674  isotrALT 4875  tfrlem1 5119  mapxpen 5589  fzrevral 7698  bccl2 8223  smores 16446

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-4 1319  ax-5o 1321

This theorem depends on definitions:  df-bi 164  df-ral 2109

Copyright terms: Public domain