Theorem pm10.52 37586

Description: Theorem *10.52 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
pm10.52	⊢ (∃𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ 𝜓))

Distinct variable group:   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem pm10.52

Step	Hyp	Ref	Expression
1		19.23v 1889	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))
2		pm5.5 350	. 2 ⊢ (∃𝑥𝜑 → ((∃𝑥𝜑 → 𝜓) ↔ 𝜓))
3	1, 2	syl5bb 271	1 ⊢ (∃𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875

This theorem depends on definitions:  df-bi 196  df-ex 1696

This theorem is referenced by: (None)