Theorem ralim 3086

Description: Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997.) (Proof shortened by Wolf Lammen, 1-Dec-2019.)

Assertion

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

Proof of Theorem ralim

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓))
2	1	ral2imi 3085	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4  ∀wral 3061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811

This theorem depends on definitions:  df-bi 206  df-ral 3062

This theorem is referenced by:  r19.30OLD  3121  ralimdaa  3257  mpteqb  7014  tz7.49  8441  mptelixpg  8925  resixpfo  8926  bnd  9883  kmlem12  10152  lbzbi  12916  r19.29uz  15293  caubnd  15301  alzdvds  16259  ptclsg  23110  isucn2  23775  fusgreghash2wsp  29580  omssubadd  33287  subgrwlk  34111  dfon2lem8  34750  fvineqsneq  36281  dford3lem2  41751  neik0pk1imk0  42783  grur1cld  42976  mnuprdlem4  43019  mnurndlem1  43025