Theorem bnj1212 35096

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

Hypotheses

Ref	Expression
bnj1212.1	⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}
bnj1212.2	⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜏))

Assertion

Ref	Expression
bnj1212	⊢ (𝜃 → 𝑥 ∈ 𝐴)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜒(𝑥)   𝜃(𝑥)   𝜏(𝑥)   𝐵(𝑥)

Proof of Theorem bnj1212

Step	Hyp	Ref	Expression
1		bnj1212.1	. . 3 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}
2	1	ssrab3 4037	. 2 ⊢ 𝐵 ⊆ 𝐴
3		bnj1212.2	. . 3 ⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜏))
4	3	simp2bi 1160	. 2 ⊢ (𝜃 → 𝑥 ∈ 𝐵)
5	2, 4	bnj1213 35095	1 ⊢ (𝜃 → 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  {crab 3416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1101  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-rab 3417  df-ss 3923

This theorem is referenced by:  bnj1204  35309  bnj1296  35318  bnj1415  35335  bnj1421  35339  bnj1442  35346  bnj1452  35349  bnj1489  35353