Theorem 2alimdv 1921

Description: Deduction form of Theorem 19.20 of [Margaris] p. 90 with two quantifiers, see alim 1813. (Contributed by NM, 27-Apr-2004.)

Hypothesis

Ref	Expression
2alimdv.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
2alimdv	⊢ (𝜑 → (∀𝑥∀𝑦𝜓 → ∀𝑥∀𝑦𝜒))

Distinct variable groups:   𝜑,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem 2alimdv

Step	Hyp	Ref	Expression
1		2alimdv.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	alimdv 1919	. 2 ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑦𝜒))
3	2	alimdv 1919	1 ⊢ (𝜑 → (∀𝑥∀𝑦𝜓 → ∀𝑥∀𝑦𝜒))

Syntax hints:   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1798  ax-4 1812  ax-5 1913

This theorem is referenced by:  dfwe2  7624  tz7.48lem  8272  ss2mcls  33530  mclsax  33531  ichnfim  44916  iscnrm3lem2  46228