Theorem hbsbw 2173

Description: If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. Version of hbsb 2544 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 12-Aug-1993.) (Revised by Gino Giotto, 23-May-2024.)

Hypothesis

Ref	Expression
hbsbw.1	⊢ (𝜑 → ∀𝑧𝜑)

Assertion

Ref	Expression
hbsbw	⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

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

Proof of Theorem hbsbw

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sb 2070	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑤(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑)))
2		hbsbw.1	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧𝜑)
3	2	imim2i 16	. . . . . . 7 ⊢ ((𝑥 = 𝑤 → 𝜑) → (𝑥 = 𝑤 → ∀𝑧𝜑))
4	3	alimi 1813	. . . . . 6 ⊢ (∀𝑥(𝑥 = 𝑤 → 𝜑) → ∀𝑥(𝑥 = 𝑤 → ∀𝑧𝜑))
5		19.21v 1940	. . . . . . . 8 ⊢ (∀𝑧(𝑥 = 𝑤 → 𝜑) ↔ (𝑥 = 𝑤 → ∀𝑧𝜑))
6	5	biimpri 231	. . . . . . 7 ⊢ ((𝑥 = 𝑤 → ∀𝑧𝜑) → ∀𝑧(𝑥 = 𝑤 → 𝜑))
7	6	alimi 1813	. . . . . 6 ⊢ (∀𝑥(𝑥 = 𝑤 → ∀𝑧𝜑) → ∀𝑥∀𝑧(𝑥 = 𝑤 → 𝜑))
8		ax-11 2158	. . . . . 6 ⊢ (∀𝑥∀𝑧(𝑥 = 𝑤 → 𝜑) → ∀𝑧∀𝑥(𝑥 = 𝑤 → 𝜑))
9	4, 7, 8	3syl 18	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝑤 → 𝜑) → ∀𝑧∀𝑥(𝑥 = 𝑤 → 𝜑))
10	9	imim2i 16	. . . 4 ⊢ ((𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑)) → (𝑤 = 𝑦 → ∀𝑧∀𝑥(𝑥 = 𝑤 → 𝜑)))
11		19.21v 1940	. . . 4 ⊢ (∀𝑧(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑)) ↔ (𝑤 = 𝑦 → ∀𝑧∀𝑥(𝑥 = 𝑤 → 𝜑)))
12	10, 11	sylibr 237	. . 3 ⊢ ((𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑)) → ∀𝑧(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑)))
13	12	hbal 2171	. 2 ⊢ (∀𝑤(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑)) → ∀𝑧∀𝑤(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑)))
14	1, 13	hbxfrbi 1826	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1536  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-11 2158

This theorem depends on definitions:  df-bi 210  df-ex 1782  df-sb 2070

This theorem is referenced by:  hbab  2787  hblem  2920