Theorem hbxfreq 2717

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1742 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 2680	. 2 ⊢ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)
3		hbxfr.2	. 2 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵)
4	2, 3	hbxfrbi 1742	1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  ∀wal 1473   = wceq 1475   ∈ wcel 1977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606

This theorem is referenced by:  bnj1317  30146  bnj1441  30165  bnj1309  30344