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Theorem hbsbw 2169
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 2529 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.
StepHypRef Expression
1 df-sb 2068 . 2 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑤(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤𝜑)))
2 hbsbw.1 . . . . . . . 8 (𝜑 → ∀𝑧𝜑)
32imim2i 16 . . . . . . 7 ((𝑥 = 𝑤𝜑) → (𝑥 = 𝑤 → ∀𝑧𝜑))
43alimi 1814 . . . . . 6 (∀𝑥(𝑥 = 𝑤𝜑) → ∀𝑥(𝑥 = 𝑤 → ∀𝑧𝜑))
5 19.21v 1942 . . . . . . . 8 (∀𝑧(𝑥 = 𝑤𝜑) ↔ (𝑥 = 𝑤 → ∀𝑧𝜑))
65biimpri 227 . . . . . . 7 ((𝑥 = 𝑤 → ∀𝑧𝜑) → ∀𝑧(𝑥 = 𝑤𝜑))
76alimi 1814 . . . . . 6 (∀𝑥(𝑥 = 𝑤 → ∀𝑧𝜑) → ∀𝑥𝑧(𝑥 = 𝑤𝜑))
8 ax-11 2154 . . . . . 6 (∀𝑥𝑧(𝑥 = 𝑤𝜑) → ∀𝑧𝑥(𝑥 = 𝑤𝜑))
94, 7, 83syl 18 . . . . 5 (∀𝑥(𝑥 = 𝑤𝜑) → ∀𝑧𝑥(𝑥 = 𝑤𝜑))
109imim2i 16 . . . 4 ((𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤𝜑)) → (𝑤 = 𝑦 → ∀𝑧𝑥(𝑥 = 𝑤𝜑)))
11 19.21v 1942 . . . 4 (∀𝑧(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤𝜑)) ↔ (𝑤 = 𝑦 → ∀𝑧𝑥(𝑥 = 𝑤𝜑)))
1210, 11sylibr 233 . . 3 ((𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤𝜑)) → ∀𝑧(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤𝜑)))
1312hbal 2167 . 2 (∀𝑤(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤𝜑)) → ∀𝑧𝑤(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤𝜑)))
141, 13hbxfrbi 1827 1 ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1537  [wsb 2067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-11 2154
This theorem depends on definitions:  df-bi 206  df-ex 1783  df-sb 2068
This theorem is referenced by:  nfsbv  2324  hbab  2726  hblem  2870
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