Theorem wl-nfsbtv 37077

Description: Closed form of nfsbv 2318. (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 2063	. 2 ⊢ (∀𝑥Ⅎ𝑧𝜑 → [𝑦 / 𝑥]Ⅎ𝑧𝜑)
2		sbnf 2301	. 2 ⊢ ([𝑦 / 𝑥]Ⅎ𝑧𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑)
3	1, 2	sylib 217	1 ⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1531  Ⅎwnf 1777  [wsb 2059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-11 2146  ax-12 2166

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1774  df-nf 1778  df-sb 2060

This theorem is referenced by:  wl-sb8eutv  37079