Theorem bnj1538 31460

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

Hypothesis

Ref	Expression
bnj1538.1	⊢ 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝜑}

Assertion

Ref	Expression
bnj1538	⊢ (𝑥 ∈ 𝐴 → 𝜑)

Proof of Theorem bnj1538

Step	Hyp	Ref	Expression
1		bnj1538.1	. . 3 ⊢ 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝜑}
2	1	rabeq2i 3410	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜑))
3	2	simprbi 492	1 ⊢ (𝑥 ∈ 𝐴 → 𝜑)

Syntax hints:   → wi 4   = wceq 1656   ∈ wcel 2164  {crab 3121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-12 2220  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-tru 1660  df-ex 1879  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-rab 3126

This theorem is referenced by:  bnj1279  31621  bnj1311  31627  bnj1418  31643  bnj1312  31661  bnj1523  31674