Theorem hbsb3 2508

Description: If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. Usage of this theorem is discouraged because it depends on ax-13 2393. Check out bj-hbsb3v 37238 for a weaker version requiring fewer axioms. (Contributed by NM, 14-May-1993.) (New usage is discouraged.)

Hypothesis

Ref	Expression
hbsb3.1	⊢ (𝜑 → ∀𝑦𝜑)

Assertion

Ref	Expression
hbsb3	⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Proof of Theorem hbsb3

Step	Hyp	Ref	Expression
1		hbsb3.1	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
2	1	sbimi 2097	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑦𝜑)
3		hbsb2a 2505	. 2 ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
4	2, 3	syl 17	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1548  [wsb 2080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-10 2165  ax-12 2202  ax-13 2393

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-ex 1790  df-nf 1794  df-sb 2081

This theorem is referenced by:  nfs1  2509  axc16ALT  2510