Theorem r19.22i2 1709

Description: Inference quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90.

Hypothesis

Ref	Expression
r19.22i2.1	|- ((x e. A /\ ph) -> (x e. B /\ ps))

Assertion

Ref	Expression
r19.22i2	|- (E.x e. A ph -> E.x e. B ps)

Proof of Theorem r19.22i2

Step	Hyp	Ref	Expression
1		r19.22i2.1	. . 3 |- ((x e. A /\ ph) -> (x e. B /\ ps))
2	1	19.22i 1016	. 2 |- (E.x(x e. A /\ ph) -> E.x(x e. B /\ ps))
3		df-rex 1626	. 2 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
4		df-rex 1626	. 2 |- (E.x e. B ps <-> E.x(x e. B /\ ps))
5	2, 3, 4	3imtr4 219	1 |- (E.x e. A ph -> E.x e. B ps)

Syntax hints:   -> wi 3   /\ wa 223  E.wex 956   e. wcel 1105  E.wrex 1622

This theorem is referenced by:  pssnn 4465  xrsupexmnf 5972  xrinfmexpnf 5973  xrsupsslem 5974  xrinfmsslem 5975  supxrun 5983  btwnz 6114  ioo0t 6256  sqr2irr 6610  nmobndseqi 8307  circgrp 8573  pjnmop 10200

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-gen 955

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957  df-rex 1626

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