Theorem pm11.57 40811

Description: Theorem *11.57 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
pm11.57	⊢ (∀𝑥𝜑 ↔ ∀𝑥∀𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))

Distinct variable group:   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem pm11.57

Step	Hyp	Ref	Expression
1		nfv 1915	. . . . 5 ⊢ Ⅎ𝑦𝜑
2	1	nfal 2342	. . . 4 ⊢ Ⅎ𝑦∀𝑥𝜑
3		sp 2182	. . . . 5 ⊢ (∀𝑥𝜑 → 𝜑)
4		stdpc4 2073	. . . . 5 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
5	3, 4	jca 514	. . . 4 ⊢ (∀𝑥𝜑 → (𝜑 ∧ [𝑦 / 𝑥]𝜑))
6	2, 5	alrimi 2213	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
7	6	axc4i 2341	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
8		simpl 485	. . . 4 ⊢ ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝜑)
9	8	sps 2184	. . 3 ⊢ (∀𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝜑)
10	9	alimi 1812	. 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑) → ∀𝑥𝜑)
11	7, 10	impbii 211	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑥∀𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))

Syntax hints:   ↔ wb 208   ∧ wa 398  ∀wal 1535  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070

This theorem is referenced by: (None)