Theorem spsbce-2 37602

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 1871	. 2 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → ∃𝑥[𝑤 / 𝑦]𝜑)
2		spsbe 1871	. . 3 ⊢ ([𝑤 / 𝑦]𝜑 → ∃𝑦𝜑)
3	2	eximi 1752	. 2 ⊢ (∃𝑥[𝑤 / 𝑦]𝜑 → ∃𝑥∃𝑦𝜑)
4	1, 3	syl 17	1 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → ∃𝑥∃𝑦𝜑)

Syntax hints:   → wi 4  ∃wex 1695  [wsb 1867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-sb 1868

This theorem is referenced by: (None)