Theorem hbs1 2274

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 2161	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
2	1	nf5ri 2197	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1540  [wsb 2074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-10 2145  ax-12 2179

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-ex 1787  df-nf 1791  df-sb 2075

This theorem is referenced by:  hbab1  2724  sb5ALT  41683  2sb5ndVD  42068  sb5ALTVD  42071  2sb5ndALT  42090