Theorem hbsb2e 1728

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

Assertion

Ref	Expression
hbsb2e	⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑)

Proof of Theorem hbsb2e

Step	Hyp	Ref	Expression
1		sb4e 1726	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
2		sb2 1690	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑) → [𝑦 / 𝑥]∃𝑦𝜑)
3	2	a5i 1475	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑) → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑)
4	1, 3	syl 14	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑)

Syntax hints:   → wi 4  ∀wal 1282  ∃wex 1421  [wsb 1685

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 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-11 1437  ax-4 1440  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-sb 1686

This theorem is referenced by: (None)