Theorem ralimda 41398

Description: Deduction quantifying both antecedent and consequent. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

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

Assertion

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

Proof of Theorem ralimda

Step	Hyp	Ref	Expression
1		ralimda.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2		nfra1 3219	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜓
3	1, 2	nfan 1896	. . 3 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)
4		id 22	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜑 ∧ 𝑥 ∈ 𝐴))
5	4	adantlr 713	. . . 4 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜑 ∧ 𝑥 ∈ 𝐴))
6		rspa 3206	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝜓 ∧ 𝑥 ∈ 𝐴) → 𝜓)
7	6	adantll 712	. . . 4 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 ∈ 𝐴) → 𝜓)
8		ralimda.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))
9	5, 7, 8	sylc 65	. . 3 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 ∈ 𝐴) → 𝜒)
10	3, 9	ralrimia 41390	. 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 𝜒)
11	10	ex 415	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ∧ wa 398  Ⅎwnf 1780   ∈ wcel 2110  ∀wral 3138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-12 2172

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-ral 3143

This theorem is referenced by:  xlimmnfvlem1  42105  xlimmnfvlem2  42106  xlimpnfvlem1  42109  xlimpnfvlem2  42110