Theorem spsbce-2 45119

Description: Theorem *11.36 in [WhiteheadRussell] p. 162. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
spsbce-2	⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → ∃𝑥∃𝑦𝜑)

Proof of Theorem spsbce-2

Step	Hyp	Ref	Expression
1		spsbe 2115	. 2 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → ∃𝑥[𝑤 / 𝑦]𝜑)
2		spsbe 2115	. . 3 ⊢ ([𝑤 / 𝑦]𝜑 → ∃𝑦𝜑)
3	2	eximi 1864	. 2 ⊢ (∃𝑥[𝑤 / 𝑦]𝜑 → ∃𝑥∃𝑦𝜑)
4	1, 3	syl 18	1 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → ∃𝑥∃𝑦𝜑)

This proof depends on syntax axioms:   → wi 4  ∃wex 1808  [wsb 2095

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-sb 2096

This theorem is used by: (None)