Theorem hblemg 2873

Description: Change the free variable of a hypothesis builder. Usage of this theorem is discouraged because it depends on ax-13 2376. See hblem 2872 for a version with more disjoint variable conditions, but not requiring ax-13 2376. (Contributed by NM, 21-Jun-1993.) (Revised by Andrew Salmon, 11-Jul-2011.) (New usage is discouraged.)

Hypothesis

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

Assertion

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

Distinct variable groups:   𝑦,𝐴   𝑥,𝑧

Allowed substitution hints:   𝐴(𝑥,𝑧)

Proof of Theorem hblemg

Step	Hyp	Ref	Expression
1		hblemg.1	. . 3 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
2	1	hbsb 2528	. 2 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 → ∀𝑥[𝑧 / 𝑦]𝑦 ∈ 𝐴)
3		clelsb1 2867	. 2 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)
4	3	albii 1818	. 2 ⊢ (∀𝑥[𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ ∀𝑥 𝑧 ∈ 𝐴)
5	2, 3, 4	3imtr3i 291	1 ⊢ (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴)

Syntax hints:   → wi 4  ∀wal 1537  [wsb 2063   ∈ wcel 2107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-10 2140  ax-11 2156  ax-12 2176  ax-13 2376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clel 2815

This theorem is referenced by: (None)