Theorem wl-nfsbtv 38085

Description: Closed form of nfsbv 2363. (Contributed by Wolf Lammen, 2-May-2025.)

Assertion

Ref	Expression
wl-nfsbtv	⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

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

Proof of Theorem wl-nfsbtv

Step	Hyp	Ref	Expression
1		stdpc4 2099	. 2 ⊢ (∀𝑥Ⅎ𝑧𝜑 → [𝑦 / 𝑥]Ⅎ𝑧𝜑)
2		sbnf 2346	. 2 ⊢ ([𝑦 / 𝑥]Ⅎ𝑧𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑)
3	1, 2	sylib 220	1 ⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1559  Ⅎwnf 1804  [wsb 2091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-10 2176  ax-11 2192  ax-12 2213

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1801  df-nf 1805  df-sb 2092

This theorem is referenced by:  wl-sb8eutv  38087