Theorem bnj1538 34891

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	reqabi 3444	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜑))
3	2	simprbi 496	1 ⊢ (𝑥 ∈ 𝐴 → 𝜑)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {crab 3420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421

This theorem is referenced by:  bnj1279  35054  bnj1311  35060  bnj1418  35076  bnj1312  35094  bnj1523  35107