Theorem pm3.2 469

Description: Join antecedents with conjunction ("conjunction introduction"). Theorem *3.2 of [WhiteheadRussell] p. 111. Its associated inference is pm3.2i 470 and its associated deduction is jca 511 (and the double deduction is jcad 512). See pm3.2im 160 for a version using only implication and negation. (Contributed by NM, 5-Jan-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.)

Assertion

Ref	Expression
pm3.2	⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))

Proof of Theorem pm3.2

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜓))
2	1	ex 412	1 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))

Syntax hints:   → wi 4   ∧ wa 395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396

This theorem is referenced by:  pm3.2i  470  pm3.43i  472  jca  511  jcad  512  ancl  544  19.29  1875  19.40b  1890  sban  2086  sb1  2483  mo4  2567  axia3  2696  r19.26  3098  difrab  4272  reuss2  4280  dmcosseq  5935  dmcosseqOLD  5936  dmcosseqOLDOLD  5937  soxp  8081  suppofssd  8155  smoord  8307  xpwdomg  9502  alephexp2  10504  lediv2a  12048  ssfzo12  13687  fzoopth  13690  r19.29uz  15286  gsummoncoe1  22264  fbun  23796  fisshasheq  35328  isdrngo3  38204  cantnf2  43676  or3or  44373  pm11.71  44747  tratrb  44886  onfrALTlem3  44894  elex22VD  45188  en3lplem1VD  45192  tratrbVD  45210  undif3VD  45231  onfrALTlem3VD  45236  19.41rgVD  45251  2pm13.193VD  45252  ax6e2eqVD  45256  2uasbanhVD  45260  vk15.4jVD  45263