Theorem bnj1441g 34833

Description: First-order logic and set theory. See bnj1441 34832 for a version with more disjoint variable conditions, but not requiring ax-13 2374. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1441g.1	⊢ (𝑥 ∈ 𝐴 → ∀𝑦 𝑥 ∈ 𝐴)
bnj1441g.2	⊢ (𝜑 → ∀𝑦𝜑)

Assertion

Ref	Expression
bnj1441g	⊢ (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 bnj1441g

Step	Hyp	Ref	Expression
1		df-rab 3433	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2		bnj1441g.1	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 𝑥 ∈ 𝐴)
3		bnj1441g.2	. . . 4 ⊢ (𝜑 → ∀𝑦𝜑)
4	2, 3	hban 2298	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∀𝑦(𝑥 ∈ 𝐴 ∧ 𝜑))
5	4	hbabg 2723	. 2 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ∀𝑦 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
6	1, 5	hbxfreq 2869	1 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → ∀𝑦 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1534   ∈ wcel 2105  {cab 2711  {crab 3432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-13 2374  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433

This theorem is referenced by: (None)