Theorem ralimia 2424

Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 19-Jul-1996.)

Hypothesis

Ref	Expression
ralimia.1	|- ( x e. A -> ( ph -> ps ) )

Assertion

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

Proof of Theorem ralimia

Step	Hyp	Ref	Expression
1		ralimia.1	. . 3 |- ( x e. A -> ( ph -> ps ) )
2	1	a2i 11	. 2 |- ( ( x e. A -> ph ) -> ( x e. A -> ps ) )
3	2	ralimi2 2423	1 |- ( A. x e. A ph -> A. x e. A ps )

Syntax hints:   -> wi 4   e. wcel 1433   A.wral 2348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378

This theorem depends on definitions:  df-bi 115  df-ral 2353

This theorem is referenced by:  ralimiaa  2425  ralimi  2426  r19.12  2466  rr19.3v  2733  rr19.28v  2734  ffvresb  5349  f1mpt  5431  peano2nnnn  7021  peano5nnnn  7058  peano5nni  8042  peano2nn  8051  serif0  10189