Theorem pm11.58 45160

Description: Theorem *11.58 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
pm11.58	⊢ (∃𝑥𝜑 ↔ ∃𝑥∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))

Distinct variable group:   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem pm11.58

Step	Hyp	Ref	Expression
1		19.8a 2220	. . . . 5 ⊢ (𝜑 → ∃𝑥𝜑)
2		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑦𝜑
3	2	sb8e 2552	. . . . 5 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
4	1, 3	sylib 221	. . . 4 ⊢ (𝜑 → ∃𝑦[𝑦 / 𝑥]𝜑)
5	4	pm4.71i 569	. . 3 ⊢ (𝜑 ↔ (𝜑 ∧ ∃𝑦[𝑦 / 𝑥]𝜑))
6		19.42v 1986	. . 3 ⊢ (∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑 ∧ ∃𝑦[𝑦 / 𝑥]𝜑))
7	5, 6	bitr4i 281	. 2 ⊢ (𝜑 ↔ ∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
8	7	exbii 1881	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑥∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∃wex 1812  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-11 2195  ax-12 2216  ax-13 2406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2100

This theorem is used by: (None)