Theorem anim12 342

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 108	. 2 |- ( ( ( ph -> ps ) /\ ( ch -> th ) ) -> ( ph -> ps ) )
2		simpr 109	. 2 |- ( ( ( ph -> ps ) /\ ( ch -> th ) ) -> ( ch -> th ) )
3	1, 2	anim12d 333	1 |- ( ( ( ph -> ps ) /\ ( ch -> th ) ) -> ( ( ph /\ ch ) -> ( ps /\ th ) ) )

Syntax hints:   -> wi 4   /\ wa 103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  nfand  1561  equsexd  1722  mo23  2060  euind  2917  reuind  2935  reuss2  3407  opelopabt  4247  reusv3i  4444  rexanre  11184  2sqlem6  13750  bj-stan  13782