Theorem hblem 2891

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 2401. See hblemg 2892 for a less restrictive version requiring more axioms. (Revised by GG, 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 2207	. 2 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 → ∀𝑥[𝑧 / 𝑦]𝑦 ∈ 𝐴)
3		clelsb1 2887	. 2 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)
4	3	albii 1851	. 2 ⊢ (∀𝑥[𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ ∀𝑥 𝑧 ∈ 𝐴)
5	2, 3, 4	3imtr3i 294	1 ⊢ (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  [wsb 2098   ∈ wcel 2144

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-11 2193

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clel 2835

This theorem is used by:  bnj1311  35654