Theorem hbsb4 2039

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 1976	. 2 ⊢ ([𝑤 / 𝑥]𝜑 → ∀𝑧[𝑤 / 𝑥]𝜑)
3		sbequ 1862	. 2 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	2, 3	dvelimALT 2037	1 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1370  [wsb 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785

This theorem is referenced by:  hbsb4t  2040  dvelimf  2042