Theorem ralimdaa 3236

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-Sep-2003.) (Proof shortened by Wolf Lammen, 29-Dec-2019.)

Hypotheses

Ref	Expression
ralimdaa.1	⊢ Ⅎ𝑥𝜑
ralimdaa.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))

Assertion

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

Proof of Theorem ralimdaa

Step	Hyp	Ref	Expression
1		ralimdaa.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		ralimdaa.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))
3	2	ex 412	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))
4	1, 3	ralrimi 3233	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
5		ralim 3069	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))
6	4, 5	syl 17	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ∧ wa 395  Ⅎwnf 1783   ∈ wcel 2109  ∀wral 3044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-ral 3045

This theorem is referenced by:  ralbida  3246  eltsk2g  10680  ptcnplem  23484  poimirlem26  37613  allbutfifvre  45646  climleltrp  45647  fnlimabslt  45650  limsupub2  45783  liminflbuz2  45786  xlimmnfvlem1  45803  xlimmnfvlem2  45804  xlimpnfvlem1  45807  xlimpnfvlem2  45808  stoweidlem61  46032  stoweid  46034  fourierdlem73  46150  smflimlem2  46743