Theorem nfsb4or 2074

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 1999	. 2 ⊢ Ⅎ𝑧[𝑤 / 𝑥]𝜑
3		sbequ 1888	. 2 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	2, 3	dvelimor 2071	1 ⊢ (∀𝑧 𝑧 = 𝑦 ∨ Ⅎ𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:   ∨ wo 716  ∀wal 1396  Ⅎwnf 1509  [wsb 1810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811

This theorem is referenced by: (None)