Theorem 19.21vv 41883

Description: Compare Theorem *11.3 in [WhiteheadRussell] p. 161. Special case of theorem 19.21 of [Margaris] p. 90 with two quantifiers. See 19.21v 1943. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
19.21vv	⊢ (∀𝑥∀𝑦(𝜓 → 𝜑) ↔ (𝜓 → ∀𝑥∀𝑦𝜑))

Distinct variable groups:   𝜓,𝑥   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 19.21vv

Step	Hyp	Ref	Expression
1		19.21v 1943	. . 3 ⊢ (∀𝑦(𝜓 → 𝜑) ↔ (𝜓 → ∀𝑦𝜑))
2	1	albii 1823	. 2 ⊢ (∀𝑥∀𝑦(𝜓 → 𝜑) ↔ ∀𝑥(𝜓 → ∀𝑦𝜑))
3		19.21v 1943	. 2 ⊢ (∀𝑥(𝜓 → ∀𝑦𝜑) ↔ (𝜓 → ∀𝑥∀𝑦𝜑))
4	2, 3	bitri 274	1 ⊢ (∀𝑥∀𝑦(𝜓 → 𝜑) ↔ (𝜓 → ∀𝑥∀𝑦𝜑))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914

This theorem depends on definitions:  df-bi 206  df-ex 1784

This theorem is referenced by: (None)