Theorem hbsb4 1936

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 1871	. 2 ⊢ ([𝑤 / 𝑥]𝜑 → ∀𝑧[𝑤 / 𝑥]𝜑)
3		sbequ 1768	. 2 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	2, 3	dvelimALT 1934	1 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1287  [wsb 1692

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 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693

This theorem is referenced by:  hbsb4t  1937  dvelimf  1939