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  3091  raleqbidvvOLD  3310  unss  4156  ralunb  4163  ssin  4205  falseral0  4482  intun  4947  intprg  4948  eqrelrel  5763  relop  5817  eqoprab2bw  7462  eqoprab2b  7463  dfer2  8675  axgroth4  10792  grothprim  10794  trclfvcotr  14982  caubnd  15332  bj-gl4  36590  bj-nnfand  36744  bj-elgab  36934  wl-alanbii  37564  ax12eq  38941  ax12el  38942  alan  42661  dford4  43025  elmapintrab  43572  elinintrab  43573  ismnuprim  44290  alimp-no-surprise  49774