Theorem hblem 2869

Description: Change the free variable of a hypothesis builder. (Contributed by NM, 21-Jun-1993.) (Revised by Andrew Salmon, 11-Jul-2011.) Add disjoint variable condition to avoid ax-13 2371. See hblemg 2870 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)

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   𝑥,𝑦

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	hbsbw 2168	. 2 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 → ∀𝑥[𝑧 / 𝑦]𝑦 ∈ 𝐴)
3		clelsb1 2865	. 2 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)
4	3	albii 1820	. 2 ⊢ (∀𝑥[𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ ∀𝑥 𝑧 ∈ 𝐴)
5	2, 3, 4	3imtr3i 290	1 ⊢ (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴)

Syntax hints:   → wi 4  ∀wal 1538  [wsb 2066   ∈ wcel 2105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-11 2153

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1781  df-sb 2067  df-clel 2815

This theorem is referenced by:  bnj1311  33110