Theorem bnj1517 34389

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

Hypothesis

Ref	Expression
bnj1517.1	⊢ 𝐴 = {𝑥 ∣ (𝜑 ∧ 𝜓)}

Assertion

Ref	Expression
bnj1517	⊢ (𝑥 ∈ 𝐴 → 𝜓)

Proof of Theorem bnj1517

Step	Hyp	Ref	Expression
1		bnj1517.1	. . 3 ⊢ 𝐴 = {𝑥 ∣ (𝜑 ∧ 𝜓)}
2	1	bnj1436 34378	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ∧ 𝜓))
3	2	simprd 495	1 ⊢ (𝑥 ∈ 𝐴 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {cab 2703

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-12 2163  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804

This theorem is referenced by:  bnj1286  34558  bnj1450  34589  bnj1501  34606