Theorem nfsb4or 2009

Description: A variable not free remains so after substitution with a distinct variable. (Contributed by Jim Kingdon, 11-May-2018.)

Hypothesis

Ref	Expression
nfsb4or.1	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
nfsb4or	⊢ (∀𝑧 𝑧 = 𝑦 ∨ Ⅎ𝑧[𝑦 / 𝑥]𝜑)

Proof of Theorem nfsb4or

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfsb4or.1	. . 3 ⊢ Ⅎ𝑧𝜑
2	1	nfsb 1934	. 2 ⊢ Ⅎ𝑧[𝑤 / 𝑥]𝜑
3		sbequ 1828	. 2 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	2, 3	dvelimor 2006	1 ⊢ (∀𝑧 𝑧 = 𝑦 ∨ Ⅎ𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:   ∨ wo 698  ∀wal 1341  Ⅎwnf 1448  [wsb 1750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751

This theorem is referenced by: (None)