Theorem bnj1517 32122

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 32111	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ∧ 𝜓))
3	2	simprd 498	1 ⊢ (𝑥 ∈ 𝐴 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {cab 2799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1540  df-ex 1781  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893

This theorem is referenced by:  bnj1286  32291  bnj1450  32322  bnj1501  32339