Theorem r19.20da 1708

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

Hypotheses

Ref	Expression
r19.20da.1	|- (ph -> A.xph)
r19.20da.2	|- ((ph /\ x e. A) -> (ps -> ch))

Assertion

Ref	Expression
r19.20da	|- (ph -> (A.x e. A ps -> A.x e. A ch))

Proof of Theorem r19.20da

Step	Hyp	Ref	Expression
1		r19.20da.1	. . 3 |- (ph -> A.xph)
2		r19.20da.2	. . . . 5 |- ((ph /\ x e. A) -> (ps -> ch))
3	2	ex 373	. . . 4 |- (ph -> (x e. A -> (ps -> ch)))
4	3	a2d 13	. . 3 |- (ph -> ((x e. A -> ps) -> (x e. A -> ch)))
5	1, 4	19.20d 996	. 2 |- (ph -> (A.x(x e. A -> ps) -> A.x(x e. A -> ch)))
6		df-ral 1649	. 2 |- (A.x e. A ps <-> A.x(x e. A -> ps))
7		df-ral 1649	. 2 |- (A.x e. A ch <-> A.x(x e. A -> ch))
8	5, 6, 7	3imtr4g 553	1 |- (ph -> (A.x e. A ps -> A.x e. A ch))

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   e. wcel 958  A.wral 1645

This theorem is referenced by:  r19.20dva 1709  uniiunlem 2132  fopab2 3823  clm4le 7081

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-4 973  ax-5o 975

This theorem depends on definitions:  df-bi 147  df-an 225  df-ral 1649

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