Theorem hbalw 2050

Description: Weak version of hbal 2168. Uses only Tarski's FOL axiom schemes. Unlike hbal 2168, this theorem requires that 𝑥 and 𝑦 be distinct, i.e., not be bundled. (Contributed by NM, 19-Apr-2017.)

Hypotheses

Ref	Expression
hbalw.1	⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))
hbalw.2	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
hbalw	⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)

Distinct variable groups:   𝑥,𝑧   𝑥,𝑦   𝜑,𝑧   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦,𝑧)

Proof of Theorem hbalw

Step	Hyp	Ref	Expression
1		hbalw.2	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2	1	alimi 1811	. 2 ⊢ (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
3		hbalw.1	. . 3 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))
4	3	alcomimw 2043	. 2 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
5	2, 4	syl 17	1 ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780

This theorem is referenced by: (None)