Theorem hblem 2458

Description: Change the free variable of a hypothesis builder. Lemma for nfcrii 2483. (Contributed by NM, 5-Aug-1993.) (Revised by Andrew Salmon, 11-Jul-2011.)

Hypothesis

Ref	Expression
hblem.1	⊢ (y ∈ A → ∀x y ∈ A)

Assertion

Ref	Expression
hblem	⊢ (z ∈ A → ∀x z ∈ A)

Distinct variable groups:   y,A   x,z

Allowed substitution hints:   A(x, z)

Proof of Theorem hblem

Step	Hyp	Ref	Expression
1		hblem.1	. . 3 ⊢ (y ∈ A → ∀x y ∈ A)
2	1	hbsb 2110	. 2 ⊢ ([z / y]y ∈ A → ∀x[z / y]y ∈ A)
3		clelsb1 2455	. 2 ⊢ ([z / y]y ∈ A ↔ z ∈ A)
4	3	albii 1566	. 2 ⊢ (∀x[z / y]y ∈ A ↔ ∀x z ∈ A)
5	2, 3, 4	3imtr3i 256	1 ⊢ (z ∈ A → ∀x z ∈ A)

This proof depends on syntax axioms:   → wi 4  ∀wal 1540  [wsb 1648   ∈ wcel 1710

This proof depends on 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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349

This theorem is used by:  nfcrii  2483