Theorem pm11.59 44415

Description: Theorem *11.59 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
pm11.59	⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑦∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜓 ∧ [𝑦 / 𝑥]𝜓)))

Distinct variable groups:   𝜑,𝑦   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem pm11.59

Step	Hyp	Ref	Expression
1		nfv 1913	. . 3 ⊢ Ⅎ𝑦(𝜑 → 𝜓)
2	1	nfal 2322	. 2 ⊢ Ⅎ𝑦∀𝑥(𝜑 → 𝜓)
3		sp 2182	. . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → 𝜓))
4		spsbim 2071	. . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
5	3, 4	anim12d 609	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜓 ∧ [𝑦 / 𝑥]𝜓)))
6	5	axc4i 2321	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜓 ∧ [𝑦 / 𝑥]𝜓)))
7	2, 6	alrimi 2212	1 ⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑦∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜓 ∧ [𝑦 / 𝑥]𝜓)))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537  [wsb 2063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-10 2140  ax-11 2156  ax-12 2176

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1779  df-nf 1783  df-sb 2064

This theorem is referenced by: (None)