Theorem ralimia 2382

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 2381	1 |- ( A. x e. A ph -> A. x e. A ps )

Syntax hints:   -> wi 4   e. wcel 1393   A.wral 2306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338

This theorem depends on definitions:  df-bi 110  df-ral 2311

This theorem is referenced by:  ralimiaa  2383  ralimi  2384  r19.12  2422  rr19.3v  2682  rr19.28v  2683  ffvresb  5328  f1mpt  5410  peano2nnnn  6929  peano5nnnn  6966  peano5nni  7917  peano2nn  7926  serif0  9871