Theorem albi 1818

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 215	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
2	1	al2imi 1815	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
3		biimpr 220	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
4	3	al2imi 1815	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜓 → ∀𝑥𝜑))
5	2, 4	impbid 212	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 206  ∀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

This theorem is referenced by:  albii  1819  nfbiit  1851  albidh  1866  19.16  2226  19.17  2227  equvel  2454  eqeq1d  2731  raleqbidvvOLD  3308  rmoeq1  3387  elabgt  3638  ralss  4021  intmin4  4941  dfiin2g  4996  eunex  5345  bj-2albi  36601  bj-hbxfrbi  36618  bj-pm11.53vw  36764  bj-sblem  36832  wl-aleq  37523  wl-sb8ft  37538  2albi  44367  ralbidar  44434  trsbcVD  44866  sbcssgVD  44872  ichal  47464