Theorem bnj1538 35213

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 3446	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜑))
3	2	simprbi 502	1 ⊢ (𝑥 ∈ 𝐴 → 𝜑)

Syntax hints:   → wi 4   = wceq 1568   ∈ wcel 2150  {crab 3423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-12 2220  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-rab 3424

This theorem is referenced by:  bnj1279  35376  bnj1311  35382  bnj1418  35398  bnj1312  35416  bnj1523  35429