Theorem hbxfreq 2311

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1494 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 2271	. 2 ⊢ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)
3		hbxfr.2	. 2 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵)
4	2, 3	hbxfrbi 1494	1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  ∀wal 1370   = wceq 1372   ∈ wcel 2175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-17 1548  ax-ial 1556  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-cleq 2197  df-clel 2200

This theorem is referenced by: (None)