Theorem hbsb2a 1687

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 1682	. 2 ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
2		sb2 1650	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)
3	2	a5i 1435	. 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 1241  [wsb 1645

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 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-11 1397  ax-4 1400  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-sb 1646

This theorem is referenced by:  hbsb3  1689