Theorem bnj1517 30920

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 30910	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ∧ 𝜓))
3	2	simprd 479	1 ⊢ (𝑥 ∈ 𝐴 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618

This theorem is referenced by:  bnj1286  31087  bnj1450  31118  bnj1501  31135