Theorem albi 1813

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 217	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
2	1	al2imi 1810	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
3		biimpr 222	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
4	3	al2imi 1810	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜓 → ∀𝑥𝜑))
5	2, 4	impbid 214	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804

This theorem depends on definitions:  df-bi 209

This theorem is referenced by:  albii  1814  nfbiit  1845  albidh  1861  19.16  2220  19.17  2221  equvel  2473  eqeq1d  2821  intmin4  4896  dfiin2g  4948  eunex  5281  bj-2albi  33940  bj-hbxfrbi  33956  bj-sblem  34161  wl-aleq  34767  2albi  40700  ralbidar  40767  trsbcVD  41201  sbcssgVD  41207  ichal  43617