Theorem ralbi 3169

Description: Distribute a restricted universal quantifier over a biconditional. Restricted quantification version of albi 1819. (Contributed by NM, 6-Oct-2003.) Reduce axiom usage. (Revised by Wolf Lammen, 17-Jun-2023.)

Assertion

Ref	Expression
ralbi	⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓))

Proof of Theorem ralbi

Step	Hyp	Ref	Expression
1		biimp 217	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
2	1	ral2imi 3158	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓))
3		biimpr 222	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
4	3	ral2imi 3158	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜑))
5	2, 4	impbid 214	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ↔ wb 208  ∀wral 3140

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 209  df-ral 3145

This theorem is referenced by:  uniiunlem  4063  iineq2  4941  reusv2lem5  5305  ralrnmptw  6862  ralrnmpt  6864  f1mpt  7021  mpo2eqb  7285  ralrnmpo  7291  rankonidlem  9259  acni2  9474  kmlem8  9585  kmlem13  9590  fimaxre3  11589  cau3lem  14716  rlim2  14855  rlim0  14867  rlim0lt  14868  catpropd  16981  funcres2b  17169  ulmss  24987  lgamgulmlem6  25613  colinearalg  26698  axpasch  26729  axcontlem2  26753  axcontlem4  26755  axcontlem7  26758  axcontlem8  26759  neibastop3  33712  bj-0int  34395  ralbi12f  35440  iineq12f  35444  pmapglbx  36907  ordelordALTVD  41208