Theorem prth 336

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
prth	|- ( ( ( ph -> ps ) /\ ( ch -> th ) ) -> ( ( ph /\ ch ) -> ( ps /\ th ) ) )

Proof of Theorem prth

Step	Hyp	Ref	Expression
1		simpl 107	. 2 |- ( ( ( ph -> ps ) /\ ( ch -> th ) ) -> ( ph -> ps ) )
2		simpr 108	. 2 |- ( ( ( ph -> ps ) /\ ( ch -> th ) ) -> ( ch -> th ) )
3	1, 2	anim12d 328	1 |- ( ( ( ph -> ps ) /\ ( ch -> th ) ) -> ( ( ph /\ ch ) -> ( ps /\ th ) ) )

Syntax hints:   -> wi 4   /\ wa 102

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

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  nfand  1500  equsexd  1657  mo23  1982  euind  2779  reuind  2795  reuss2  3244  opelopabt  4017  reusv3i  4209  rexanre  10106