Theorem albi 1914

Description: Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 24-Jan-1993.)

Assertion

Ref	Expression
albi	⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))

Proof of Theorem albi

Step	Hyp	Ref	Expression
1		biimp 207	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
2	1	al2imi 1911	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
3		biimpr 212	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
4	3	al2imi 1911	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜓 → ∀𝑥𝜑))
5	2, 4	impbid 204	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905

This theorem depends on definitions:  df-bi 199

This theorem is referenced by:  albii  1915  nfbiit  1947  albidh  1964  19.16  2260  19.17  2261  equvel  2463  mobi  2592  mobiOLD  2593  eqeq1d  2801  intmin4  4696  dfiin2g  4743  eunex  5059  bj-2albi  33102  bj-hbxfrbi  33113  bj-eunex  33295  wl-aleq  33812  2albi  39359  ralbidar  39429  trsbcVD  39873  sbcssgVD  39879