Theorem ralimralim 39253

Description: Introducing any antecedent in a restricted universal quantification. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Assertion

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

Proof of Theorem ralimralim

Step	Hyp	Ref	Expression
1		nfra1 2941	. 2 |- F/ x A. x e. A ph
2		rspa 2930	. . . 4 |- ( ( A. x e. A ph /\ x e. A ) -> ph )
3		ax-1 6	. . . 4 |- ( ph -> ( ps -> ph ) )
4	2, 3	syl 17	. . 3 |- ( ( A. x e. A ph /\ x e. A ) -> ( ps -> ph ) )
5	4	ex 450	. 2 |- ( A. x e. A ph -> ( x e. A -> ( ps -> ph ) ) )
6	1, 5	ralrimi 2957	1 |- ( A. x e. A ph -> A. x e. A ( ps -> ph ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-ral 2917

This theorem is referenced by:  infxrunb2  39584