Theorem bnj1476 35524

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1476.1	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}
bnj1476.2	⊢ (𝜓 → 𝐷 = ∅)

Assertion

Ref	Expression
bnj1476	⊢ (𝜓 → ∀𝑥 ∈ 𝐴 𝜑)

Proof of Theorem bnj1476

Step	Hyp	Ref	Expression
1		bnj1476.2	. . . 4 ⊢ (𝜓 → 𝐷 = ∅)
2		bnj1476.1	. . . . . 6 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}
3		nfrab1 3431	. . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ¬ 𝜑}
4	2, 3	nfcxfr 2920	. . . . 5 ⊢ Ⅎ𝑥𝐷
5	4	eq0f 4293	. . . 4 ⊢ (𝐷 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐷)
6	1, 5	sylib 221	. . 3 ⊢ (𝜓 → ∀𝑥 ¬ 𝑥 ∈ 𝐷)
7	2	reqabi 3434	. . . . 5 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑))
8	7	notbii 323	. . . 4 ⊢ (¬ 𝑥 ∈ 𝐷 ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑))
9		iman 407	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑))
10	8, 9	sylbb2 241	. . 3 ⊢ (¬ 𝑥 ∈ 𝐷 → (𝑥 ∈ 𝐴 → 𝜑))
11	6, 10	sylg 1855	. 2 ⊢ (𝜓 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
12	11	ralrid 3084	1 ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∀wal 1567   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412  ∅c0 4278

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rab 3413  df-dif 4047  df-nul 4279

This theorem is used by:  bnj1312  35735