Theorem rr19.28v 3346

Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the nonempty class condition of r19.28zv 4066 when there is an outer quantifier. (Contributed by NM, 29-Oct-2012.)

Assertion

Ref	Expression
rr19.28v	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem rr19.28v

Step	Hyp	Ref	Expression
1		simpl 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	ralimi 2952	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∀𝑦 ∈ 𝐴 𝜑)
3		biidd 252	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜑))
4	3	rspcv 3305	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝜑 → 𝜑))
5	2, 4	syl5 34	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) → 𝜑))
6		simpr 477	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
7	6	ralimi 2952	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∀𝑦 ∈ 𝐴 𝜓)
8	7	a1i 11	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∀𝑦 ∈ 𝐴 𝜓))
9	5, 8	jcad 555	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) → (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)))
10	9	ralimia 2950	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))
11		r19.28v 3071	. . 3 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓) → ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓))
12	11	ralimi 2952	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓))
13	10, 12	impbii 199	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∈ wcel 1990  ∀wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202

This theorem is referenced by: (None)