Theorem pm11.58 44359

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 2182	. . . . 5 ⊢ (𝜑 → ∃𝑥𝜑)
2		nfv 1913	. . . . . 6 ⊢ Ⅎ𝑦𝜑
3	2	sb8e 2526	. . . . 5 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
4	1, 3	sylib 218	. . . 4 ⊢ (𝜑 → ∃𝑦[𝑦 / 𝑥]𝜑)
5	4	pm4.71i 559	. . 3 ⊢ (𝜑 ↔ (𝜑 ∧ ∃𝑦[𝑦 / 𝑥]𝜑))
6		19.42v 1953	. . 3 ⊢ (∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑 ∧ ∃𝑦[𝑦 / 𝑥]𝜑))
7	5, 6	bitr4i 278	. 2 ⊢ (𝜑 ↔ ∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
8	7	exbii 1846	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑥∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∃wex 1777  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2158  ax-12 2178  ax-13 2380

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ex 1778  df-nf 1782  df-sb 2065

This theorem is referenced by: (None)