Theorem hbsb4 1929

Description: A variable not free remains so after substitution with a distinct variable. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)

Hypothesis

Ref	Expression
hbsb4.1	⊢ (𝜑 → ∀𝑧𝜑)

Assertion

Ref	Expression
hbsb4	⊢ (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑))

Proof of Theorem hbsb4

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hbsb4.1	. . 3 ⊢ (𝜑 → ∀𝑧𝜑)
2	1	hbsb 1864	. 2 ⊢ ([𝑤 / 𝑥]𝜑 → ∀𝑧[𝑤 / 𝑥]𝜑)
3		sbequ 1761	. 2 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	2, 3	dvelimALT 1927	1 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1282  [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by:  hbsb4t  1930  dvelimf  1932