Theorem hbxfreq 2457

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1568 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 2417	. 2 ⊢ (y ∈ A ↔ y ∈ B)
3		hbxfr.2	. 2 ⊢ (y ∈ B → ∀x y ∈ B)
4	2, 3	hbxfrbi 1568	1 ⊢ (y ∈ A → ∀x y ∈ A)

Syntax hints:   → wi 4  ∀wal 1540   = wceq 1642   ∈ wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349

This theorem is referenced by:  hboprab1  5560  hboprab2  5561  hboprab3  5562  hboprab  5563