Theorem hbsbwOLD 2341

Description: Obsolete version of hbsbw 2174 as of 23-May-2024. (Contributed by NM, 12-Aug-1993.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
hbsbwOLD.1	⊢ (𝜑 → ∀𝑧𝜑)

Assertion

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

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

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

Proof of Theorem hbsbwOLD

Step	Hyp	Ref	Expression
1		hbsbwOLD.1	. . . 4 ⊢ (𝜑 → ∀𝑧𝜑)
2	1	nf5i 2148	. . 3 ⊢ Ⅎ𝑧𝜑
3	2	nfsbv 2339	. 2 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑
4	3	nf5ri 2194	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1537  [wsb 2070

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 1912  ax-6 1971  ax-7 2016  ax-10 2143  ax-11 2159  ax-12 2176

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1783  df-nf 1787  df-sb 2071

This theorem is referenced by: (None)