Theorem hbbid 1563

Description: Deduction form of bound-variable hypothesis builder hbbi 1536. (Contributed by NM, 1-Jan-2002.)

Hypotheses

Ref	Expression
hbbid.1	⊢ (𝜑 → ∀𝑥𝜑)
hbbid.2	⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
hbbid.3	⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))

Assertion

Ref	Expression
hbbid	⊢ (𝜑 → ((𝜓 ↔ 𝜒) → ∀𝑥(𝜓 ↔ 𝜒)))

Proof of Theorem hbbid

Step	Hyp	Ref	Expression
1		hbbid.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
2		hbbid.2	. . . 4 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
3		hbbid.3	. . . 4 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
4	1, 2, 3	hbimd 1561	. . 3 ⊢ (𝜑 → ((𝜓 → 𝜒) → ∀𝑥(𝜓 → 𝜒)))
5	1, 3, 2	hbimd 1561	. . 3 ⊢ (𝜑 → ((𝜒 → 𝜓) → ∀𝑥(𝜒 → 𝜓)))
6	4, 5	anim12d 333	. 2 ⊢ (𝜑 → (((𝜓 → 𝜒) ∧ (𝜒 → 𝜓)) → (∀𝑥(𝜓 → 𝜒) ∧ ∀𝑥(𝜒 → 𝜓))))
7		dfbi2 386	. 2 ⊢ ((𝜓 ↔ 𝜒) ↔ ((𝜓 → 𝜒) ∧ (𝜒 → 𝜓)))
8		albiim 1475	. 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) ↔ (∀𝑥(𝜓 → 𝜒) ∧ ∀𝑥(𝜒 → 𝜓)))
9	6, 7, 8	3imtr4g 204	1 ⊢ (𝜑 → ((𝜓 ↔ 𝜒) → ∀𝑥(𝜓 ↔ 𝜒)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-4 1498  ax-i5r 1523

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)