Theorem bnj1517 34857

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 34846	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ∧ 𝜓))
3	2	simprd 495	1 ⊢ (𝑥 ∈ 𝐴 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816

This theorem is referenced by:  bnj1286  35026  bnj1450  35057  bnj1501  35074