Theorem r19.23ai 2209

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (The proof was shortened by Andrew Salmon, 30-May-2011.)

Hypotheses

Ref	Expression
r19.23ai.1	|- (ps -> A.xps)
r19.23ai.2	|- (x e. A -> (ph -> ps))

Assertion

Ref	Expression
r19.23ai	|- (E.x e. A ph -> ps)

Proof of Theorem r19.23ai

Step	Hyp	Ref	Expression
1		r19.23ai.2	. . 3 |- (x e. A -> (ph -> ps))
2	1	rgen 2159	. 2 |- A.x e. A (ph -> ps)
3		r19.23ai.1	. . 3 |- (ps -> A.xps)
4	3	r19.23 2206	. 2 |- (A.x e. A (ph -> ps) <-> (E.x e. A ph -> ps))
5	2, 4	mpbi 206	1 |- (E.x e. A ph -> ps)

Syntax hints:   -> wi 3  A.wal 1296   e. wcel 1300  A.wral 2105  E.wrex 2106

This theorem is referenced by:  r19.23aiv 2211  tfinds 3942  tfindsOLD 3943  ordtypelem7 5690  r1val1 5769  rankuni2 5801  dfon2lem7 13855  bwt2 14960  finminlem 15367  ordtypelem7OLD 15381

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  ax-6o 1324

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-ral 2109  df-rex 2110

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