Theorem pm11.58 38907

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 2090	. . . . 5 ⊢ (𝜑 → ∃𝑥𝜑)
2		nfv 1883	. . . . . 6 ⊢ Ⅎ𝑦𝜑
3	2	sb8e 2453	. . . . 5 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
4	1, 3	sylib 208	. . . 4 ⊢ (𝜑 → ∃𝑦[𝑦 / 𝑥]𝜑)
5	4	pm4.71i 665	. . 3 ⊢ (𝜑 ↔ (𝜑 ∧ ∃𝑦[𝑦 / 𝑥]𝜑))
6		19.42v 1921	. . 3 ⊢ (∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑 ∧ ∃𝑦[𝑦 / 𝑥]𝜑))
7	5, 6	bitr4i 267	. 2 ⊢ (𝜑 ↔ ∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
8	7	exbii 1814	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑥∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))

Syntax hints:   ↔ wb 196   ∧ wa 383  ∃wex 1744  [wsb 1937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-sb 1938

This theorem is referenced by: (None)