Theorem albi 1819

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 1816	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
3		biimpr 220	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
4	3	al2imi 1816	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜓 → ∀𝑥𝜑))
5	2, 4	impbid 212	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 207

This theorem is referenced by:  albii  1820  nfbiit  1852  albidh  1867  19.16  2228  19.17  2229  equvel  2456  eqeq1d  2733  raleqbidvvOLD  3301  rmoeq1  3377  elabgt  3627  ralss  4009  intmin4  4927  dfiin2g  4981  eunex  5328  bj-2albi  36646  bj-hbxfrbi  36663  bj-pm11.53vw  36809  bj-sblem  36877  wl-aleq  37568  wl-sb8ft  37583  2albi  44410  ralbidar  44476  trsbcVD  44908  sbcssgVD  44914  ichal  47496