Theorem ralimdaa 3239

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	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))
3	1, 2	ralrimia 3237	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
4		ralim 3078	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))
5	3, 4	syl 17	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ∧ wa 395  Ⅎwnf 1785   ∈ wcel 2114  ∀wral 3052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-nf 1786  df-ral 3053

This theorem is referenced by:  ralbida  3249  eltsk2g  10668  ptcnplem  23599  poimirlem26  37984  allbutfifvre  46124  climleltrp  46125  fnlimabslt  46128  limsupub2  46261  liminflbuz2  46264  xlimmnfvlem1  46281  xlimmnfvlem2  46282  xlimpnfvlem1  46285  xlimpnfvlem2  46286  stoweidlem61  46510  stoweid  46512  fourierdlem73  46628  smflimlem2  47221