Theorem hbsb3 1730

Description: If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. (Contributed by NM, 5-Aug-1993.)

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 1688	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑦𝜑)
3		hbsb2a 1728	. 2 ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
4	2, 3	syl 14	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1283  [wsb 1686

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-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-11 1438  ax-4 1441  ax-i9 1464  ax-ial 1468

This theorem depends on definitions:  df-bi 115  df-sb 1687

This theorem is referenced by:  nfs1  1731  sbcof2  1732  ax16  1735  sb8h  1776  sb8eh  1777  ax16ALT  1781