Theorem 3anim123d 1353

Description: Deduction joining 3 implications to form implication of conjunctions. (Contributed by NM, 24-Feb-2005.)

Hypotheses

Ref	Expression
3anim123d.1	|- ( ph -> ( ps -> ch ) )
3anim123d.2	|- ( ph -> ( th -> ta ) )
3anim123d.3	|- ( ph -> ( et -> ze ) )

Assertion

Ref	Expression
3anim123d	|- ( ph -> ( ( ps /\ th /\ et ) -> ( ch /\ ta /\ ze ) ) )

Proof of Theorem 3anim123d

Step	Hyp	Ref	Expression
1		3anim123d.1	. . . 4 |- ( ph -> ( ps -> ch ) )
2		3anim123d.2	. . . 4 |- ( ph -> ( th -> ta ) )
3	1, 2	anim12d 335	. . 3 |- ( ph -> ( ( ps /\ th ) -> ( ch /\ ta ) ) )
4		3anim123d.3	. . 3 |- ( ph -> ( et -> ze ) )
5	3, 4	anim12d 335	. 2 |- ( ph -> ( ( ( ps /\ th ) /\ et ) -> ( ( ch /\ ta ) /\ ze ) ) )
6		df-3an 1004	. 2 |- ( ( ps /\ th /\ et ) <-> ( ( ps /\ th ) /\ et ) )
7		df-3an 1004	. 2 |- ( ( ch /\ ta /\ ze ) <-> ( ( ch /\ ta ) /\ ze ) )
8	5, 6, 7	3imtr4g 205	1 |- ( ph -> ( ( ps /\ th /\ et ) -> ( ch /\ ta /\ ze ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002

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  df-3an 1004

This theorem is referenced by:  hb3and  1536  pofun  4403  soss  4405  wessep  4670  isopolem  5952  isosolem  5954  issmo2  6441  smores  6444  issubmnd  13490  issubg2m  13741  issubrng2  14189  issubrg2  14220  rnglidlmsgrp  14476  rnglidlrng  14477  sslm  14936