Theorem 19.26 1870

Description: Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 147. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)

Assertion

Ref	Expression
19.26	⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))

Proof of Theorem 19.26

Step	Hyp	Ref	Expression
1		simpl 482	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	alimi 1811	. . 3 ⊢ (∀𝑥(𝜑 ∧ 𝜓) → ∀𝑥𝜑)
3		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
4	3	alimi 1811	. . 3 ⊢ (∀𝑥(𝜑 ∧ 𝜓) → ∀𝑥𝜓)
5	2, 4	jca 511	. 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) → (∀𝑥𝜑 ∧ ∀𝑥𝜓))
6		id 22	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜓))
7	6	alanimi 1816	. 2 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥(𝜑 ∧ 𝜓))
8	5, 7	impbii 209	1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

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

This theorem is referenced by:  19.26-2  1871  19.26-3an  1872  19.43OLD  1883  albiim  1889  2albiim  1890  19.27v  1995  19.28v  1996  19.27  2228  19.28  2229  r19.26m  3090  raleqbidvvOLD  3305  unss  4149  ralunb  4156  ssin  4198  falseral0  4475  intun  4940  intprg  4941  eqrelrel  5751  relop  5804  eqoprab2bw  7439  eqoprab2b  7440  dfer2  8649  axgroth4  10761  grothprim  10763  trclfvcotr  14951  caubnd  15301  bj-gl4  36556  bj-nnfand  36710  bj-elgab  36900  wl-alanbii  37530  ax12eq  38907  ax12el  38908  alan  42627  dford4  42991  elmapintrab  43538  elinintrab  43539  ismnuprim  44256  alimp-no-surprise  49743