Theorem r19.21ai 1258

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

Hypotheses

Ref	Expression
r19.21ai.1	|- (ph -> A.xph)
r19.21ai.2	|- (ph -> (x e. A -> ps))

Assertion

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

Proof of Theorem r19.21ai

Step	Hyp	Ref	Expression
1		r19.21ai.1	. . 3 |- (ph -> A.xph)
2		r19.21ai.2	. . 3 |- (ph -> (x e. A -> ps))
3	1, 2	19.21ai 740	. 2 |- (ph -> A.x(x e. A -> ps))
4		df-ral 1205	. 2 |- (A.x e. A ps <-> A.x(x e. A -> ps))
5	3, 4	sylibr 175	1 |- (ph -> A.x e. A ps)

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

This theorem is referenced by:  r19.21aiv 1259  r19.22d 1276  r19.12 1281  zfrep6 2744  fnopabg 2745  isotrALT 2936  tfr3 2964  mapxpen 3390  aceq6b 3565  ac6lem 3575

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-gen 677

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

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