Theorem ralimia 3081

Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 19-Jul-1996.)

Hypothesis

Ref	Expression
ralimia.1	⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))

Assertion

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

Proof of Theorem ralimia

Step	Hyp	Ref	Expression
1		ralimia.1	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
2	1	a2i 14	. 2 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → (𝑥 ∈ 𝐴 → 𝜓))
3	2	ralimi2 3079	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∈ wcel 2107  ∀wral 3062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-ral 3063

This theorem is referenced by:  ralimiaa  3083  ralimi  3084  rr19.3v  3657  rr19.28v  3658  exfo  7104  ffvresb  7121  f1mpt  7257  weniso  7348  xpord2indlem  8130  ixpf  8911  ixpiunwdom  9582  tz9.12lem3  9781  dfac2a  10121  kmlem12  10153  axdc2lem  10440  ac6c4  10473  brdom6disj  10524  konigthlem  10560  arch  12466  cshw1  14769  serf0  15624  symgextfo  19285  baspartn  22449  ptcnplem  23117  spanuni  30785  lnopunilem1  31251  phpreu  36461  finixpnum  36462  poimirlem26  36503  indexa  36590  heiborlem5  36672  rngmgmbs4  36788  mzpincl  41458  dfac11  41790  mnurndlem1  43026  natlocalincr  45577  stgoldbwt  46431  2zrngnmlid2  46803