Theorem bnj1212 34813

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 4082	. 2 ⊢ 𝐵 ⊆ 𝐴
3		bnj1212.2	. . 3 ⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜏))
4	3	simp2bi 1147	. 2 ⊢ (𝜃 → 𝑥 ∈ 𝐵)
5	2, 4	bnj1213 34812	1 ⊢ (𝜃 → 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  {crab 3436

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-ss 3968

This theorem is referenced by:  bnj1204  35026  bnj1296  35035  bnj1415  35052  bnj1421  35056  bnj1442  35063  bnj1452  35066  bnj1489  35070