Theorem bnj1476 34609

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 3438	. . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ¬ 𝜑}
4	2, 3	nfcxfr 2889	. . . . 5 ⊢ Ⅎ𝑥𝐷
5	4	eq0f 4340	. . . 4 ⊢ (𝐷 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐷)
6	1, 5	sylib 217	. . 3 ⊢ (𝜓 → ∀𝑥 ¬ 𝑥 ∈ 𝐷)
7	2	reqabi 3441	. . . . 5 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑))
8	7	notbii 319	. . . 4 ⊢ (¬ 𝑥 ∈ 𝐷 ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑))
9		iman 400	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑))
10	8, 9	sylbb2 237	. . 3 ⊢ (¬ 𝑥 ∈ 𝐷 → (𝑥 ∈ 𝐴 → 𝜑))
11	6, 10	sylg 1817	. 2 ⊢ (𝜓 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
12	11	bnj1142 34551	1 ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394  ∀wal 1531   = wceq 1533   ∈ wcel 2098  ∀wral 3050  {crab 3418  ∅c0 4322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rab 3419  df-dif 3947  df-nul 4323

This theorem is referenced by:  bnj1312  34820