Theorem 2alimdv 1947

Description: Deduction form of Theorem 19.20 of [Margaris] p. 90 with two quantifiers, see alim 1839. (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 1945	. 2 ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑦𝜒))
3	2	alimdv 1945	1 ⊢ (𝜑 → (∀𝑥∀𝑦𝜓 → ∀𝑥∀𝑦𝜒))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1824  ax-4 1838  ax-5 1939

This theorem is used by:  dfwe2  7771  tz7.48lem  8426  ss2mcls  36068  mclsax  36069  ichnfim  48241  iscnrm3lem2  49741