Theorem hbs1 2275

Description: The setvar 𝑥 is not free in [𝑦 / 𝑥]𝜑 when 𝑥 and 𝑦 are distinct. (Contributed by NM, 26-May-1993.)

Assertion

Ref	Expression
hbs1	⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem hbs1

Step	Hyp	Ref	Expression
1		nfs1v 2274	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
2	1	nf5ri 2219	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1629  [wsb 2049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-12 2203

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-ex 1853  df-nf 1858  df-sb 2050

This theorem is referenced by:  nfs1vOLD  2578  hbab1  2760  sb5ALT  39256  2sb5ndVD  39668  sb5ALTVD  39671  2sb5ndALT  39690