Theorem ralimdaa 2450

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90.

Hypotheses

Ref	Expression
ralimdaa.1	|- (ph -> A.xph)
ralimdaa.2	|- ((ph /\ x e. A) -> (ps -> ch))

Assertion

Ref	Expression
ralimdaa	|- (ph -> (A.x e. A ps -> A.x e. A ch))

Proof of Theorem ralimdaa

Step	Hyp	Ref	Expression
1		ralimdaa.1	. . 3 |- (ph -> A.xph)
2		ralimdaa.2	. . . . 5 |- ((ph /\ x e. A) -> (ps -> ch))
3	2	ex 494	. . . 4 |- (ph -> (x e. A -> (ps -> ch)))
4	3	a2d 19	. . 3 |- (ph -> ((x e. A -> ps) -> (x e. A -> ch)))
5	1, 4	alimd 1661	. 2 |- (ph -> (A.x(x e. A -> ps) -> A.x(x e. A -> ch)))
6		df-ral 2389	. 2 |- (A.x e. A ps <-> A.x(x e. A -> ps))
7		df-ral 2389	. 2 |- (A.x e. A ch <-> A.x(x e. A -> ch))
8	5, 6, 7	3imtr4g 333	1 |- (ph -> (A.x e. A ps -> A.x e. A ch))

Syntax hints:   -> wi 3   /\ wa 433  A.wal 1613   e. wcel 1617  A.wral 2385

This theorem is referenced by:  ralimdva 2451  uniiunlem 2950  fopab2 4927  clm4lei 8853  fopab2g 15480  npincppr 15501  fprodneg 15737  taralt 16282  mettrifi 16932

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1622  ax-4 1637  ax-5o 1639

This theorem depends on definitions:  df-bi 232  df-an 435  df-ral 2389

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