Theorem pm10.52 41983

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 1945	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))
2		pm5.5 362	. 2 ⊢ (∃𝑥𝜑 → ((∃𝑥𝜑 → 𝜓) ↔ 𝜓))
3	1, 2	syl5bb 283	1 ⊢ (∃𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by: (None)