Theorem anim12 344

Description: Theorem *3.47 of [WhiteheadRussell] p. 113. It was proved by Leibniz, and it evidently pleased him enough to call it 'praeclarum theorema' (splendid theorem). (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)

Assertion

Ref	Expression
anim12	|- ( ( ( ph -> ps ) /\ ( ch -> th ) ) -> ( ( ph /\ ch ) -> ( ps /\ th ) ) )

Proof of Theorem anim12

Step	Hyp	Ref	Expression
1		simpl 109	. 2 |- ( ( ( ph -> ps ) /\ ( ch -> th ) ) -> ( ph -> ps ) )
2		simpr 110	. 2 |- ( ( ( ph -> ps ) /\ ( ch -> th ) ) -> ( ch -> th ) )
3	1, 2	anim12d 335	1 |- ( ( ( ph -> ps ) /\ ( ch -> th ) ) -> ( ( ph /\ ch ) -> ( ps /\ th ) ) )

Syntax hints:   -> wi 4   /\ wa 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  nfand  1566  equsexd  1727  mo23  2065  euind  2922  reuind  2940  reuss2  3413  opelopabt  4256  reusv3i  4453  rexanre  11197  2sqlem6  14027  bj-stan  14059