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  3658  rr19.28v  3659  exfo  7107  ffvresb  7124  f1mpt  7260  weniso  7351  xpord2indlem  8133  ixpf  8914  ixpiunwdom  9585  tz9.12lem3  9784  dfac2a  10124  kmlem12  10156  axdc2lem  10443  ac6c4  10476  brdom6disj  10527  konigthlem  10563  arch  12469  cshw1  14772  serf0  15627  symgextfo  19290  baspartn  22457  ptcnplem  23125  spanuni  30797  lnopunilem1  31263  phpreu  36472  finixpnum  36473  poimirlem26  36514  indexa  36601  heiborlem5  36683  rngmgmbs4  36799  mzpincl  41472  dfac11  41804  mnurndlem1  43040  natlocalincr  45590  stgoldbwt  46444  2zrngnmlid2  46849