Theorem hbsb2a 1665

Description: Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)

Assertion

Ref	Expression
hbsb2a	⊢ ([y / x]∀yφ → ∀x[y / x]φ)

Proof of Theorem hbsb2a

Step	Hyp	Ref	Expression
1		sb4a 1660	. 2 ⊢ ([y / x]∀yφ → ∀x(x = y → φ))
2		sb2 1628	. . 3 ⊢ (∀x(x = y → φ) → [y / x]φ)
3	2	a5i 1413	. 2 ⊢ (∀x(x = y → φ) → ∀x[y / x]φ)
4	1, 3	syl 14	1 ⊢ ([y / x]∀yφ → ∀x[y / x]φ)

Syntax hints:   → wi 4  ∀wal 1224  [wsb 1623

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1312  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-11 1374  ax-4 1377  ax-i9 1400  ax-ial 1405

This theorem depends on definitions:  df-bi 110  df-sb 1624

This theorem is referenced by:  hbsb3  1667