Theorem 2albi 1110

Description: Split a biconditional and distribute 2 quantifiers.

Assertion

Ref	Expression
2albi	⊢ (∀x∀y(φ ↔ ψ) ↔ (∀x∀y(φ → ψ) ⋀ ∀x∀y(ψ → φ)))

Proof of Theorem 2albi

Step	Hyp	Ref	Expression
1		albi 1109	. . 3 ⊢ (∀y(φ ↔ ψ) ↔ (∀y(φ → ψ) ⋀ ∀y(ψ → φ)))
2	1	albii 1001	. 2 ⊢ (∀x∀y(φ ↔ ψ) ↔ ∀x(∀y(φ → ψ) ⋀ ∀y(ψ → φ)))
3		19.26 1069	. 2 ⊢ (∀x(∀y(φ → ψ) ⋀ ∀y(ψ → φ)) ↔ (∀x∀y(φ → ψ) ⋀ ∀x∀y(ψ → φ)))
4	2, 3	bitr 173	1 ⊢ (∀x∀y(φ ↔ ψ) ↔ (∀x∀y(φ → ψ) ⋀ ∀x∀y(ψ → φ)))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 956

This theorem is referenced by:  2eu6 1457  eqrel 3256

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-4 975  ax-5o 977

This theorem depends on definitions:  df-bi 147  df-an 225

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