Theorem nfsbvOLD 2349

Description: Obsolete version of nfsbv 2348 as of 13-Aug-2023. (Contributed by Wolf Lammen, 7-Feb-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
nfsbv.nf	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
nfsbvOLD	⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Distinct variable group:   𝑥,𝑧,𝑦

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

Proof of Theorem nfsbvOLD

Step	Hyp	Ref	Expression
1		sb6 2092	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
2		nfv 1914	. . . 4 ⊢ Ⅎ𝑧 𝑥 = 𝑦
3		nfsbv.nf	. . . 4 ⊢ Ⅎ𝑧𝜑
4	2, 3	nfim 1896	. . 3 ⊢ Ⅎ𝑧(𝑥 = 𝑦 → 𝜑)
5	4	nfal 2341	. 2 ⊢ Ⅎ𝑧∀𝑥(𝑥 = 𝑦 → 𝜑)
6	1, 5	nfxfr 1852	1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Syntax hints:   → wi 4  ∀wal 1534  Ⅎwnf 1783  [wsb 2068

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 1969  ax-7 2014  ax-10 2144  ax-11 2160  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-nf 1784  df-sb 2069

This theorem is referenced by: (None)