Theorem hbxfreq 2942

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1825 for equivalence version. (Contributed by NM, 21-Aug-2007.)

Hypotheses

Ref	Expression
hbxfr.1	⊢ 𝐴 = 𝐵
hbxfr.2	⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵)

Assertion

Ref	Expression
hbxfreq	⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Proof of Theorem hbxfreq

Step	Hyp	Ref	Expression
1		hbxfr.1	. . 3 ⊢ 𝐴 = 𝐵
2	1	eleq2i 2904	. 2 ⊢ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)
3		hbxfr.2	. 2 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵)
4	2, 3	hbxfrbi 1825	1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  ∀wal 1535   = wceq 1537   ∈ wcel 2114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-cleq 2814  df-clel 2893

This theorem is referenced by:  bnj1317  32093  bnj1441  32112  bnj1441g  32113  bnj1309  32294