Theorem rr19.28v 2947

Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. (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 109	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	ralimi 2596	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∀𝑦 ∈ 𝐴 𝜑)
3		biidd 172	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜑))
4	3	rspcv 2907	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝜑 → 𝜑))
5	2, 4	syl5 32	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) → 𝜑))
6		simpr 110	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
7	6	ralimi 2596	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∀𝑦 ∈ 𝐴 𝜓)
8	7	a1i 9	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∀𝑦 ∈ 𝐴 𝜓))
9	5, 8	jcad 307	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) → (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)))
10	9	ralimia 2594	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))
11		r19.28av 2670	. . 3 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓) → ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓))
12	11	ralimi 2596	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓))
13	10, 12	impbii 126	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2202  ∀wral 2511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805

This theorem is referenced by: (None)