Theorem bnj1230 35160

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

Hypothesis

Ref	Expression
bnj1230.1	⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}

Assertion

Ref	Expression
bnj1230	⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem bnj1230

Step	Hyp	Ref	Expression
1		bnj1230.1	. . 3 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}
2		nfrab1 3443	. . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑}
3	1, 2	nfcxfr 2930	. 2 ⊢ Ⅎ𝑥𝐵
4	3	nfcrii 2927	1 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵)

Syntax hints:   → wi 4  ∀wal 1566   = wceq 1568   ∈ wcel 2150  {crab 3423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-12 2220  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1808  df-nf 1812  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-rab 3424

This theorem is referenced by:  bnj1312  35416