Theorem bnj1441 31457

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

Hypotheses

Ref	Expression
bnj1441.1	⊢ (𝑥 ∈ 𝐴 → ∀𝑦 𝑥 ∈ 𝐴)
bnj1441.2	⊢ (𝜑 → ∀𝑦𝜑)

Assertion

Ref	Expression
bnj1441	⊢ (𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → ∀𝑦 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})

Distinct variable group:   𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑥,𝑦,𝑧)

Proof of Theorem bnj1441

Step	Hyp	Ref	Expression
1		df-rab 3126	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2		bnj1441.1	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 𝑥 ∈ 𝐴)
3		bnj1441.2	. . . 4 ⊢ (𝜑 → ∀𝑦𝜑)
4	2, 3	hban 2334	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∀𝑦(𝑥 ∈ 𝐴 ∧ 𝜑))
5	4	hbab 2816	. 2 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ∀𝑦 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
6	1, 5	hbxfreq 2935	1 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → ∀𝑦 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})

Syntax hints:   → wi 4   ∧ wa 386  ∀wal 1656   ∈ wcel 2166  {cab 2811  {crab 3121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-rab 3126

This theorem is referenced by: (None)