Theorem r19.37zv 3647

Description: Restricted quantifier version of Theorem 19.37 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by Paul Chapman, 8-Oct-2007.)

Assertion

Ref	Expression
r19.37zv	⊢ (A ≠ ∅ → (∃x ∈ A (φ → ψ) ↔ (φ → ∃x ∈ A ψ)))

Distinct variable groups:   x,A   φ,x

Allowed substitution hint:   ψ(x)

Proof of Theorem r19.37zv

Step	Hyp	Ref	Expression
1		r19.3rzv 3644	. . 3 ⊢ (A ≠ ∅ → (φ ↔ ∀x ∈ A φ))
2	1	imbi1d 308	. 2 ⊢ (A ≠ ∅ → ((φ → ∃x ∈ A ψ) ↔ (∀x ∈ A φ → ∃x ∈ A ψ)))
3		r19.35 2759	. 2 ⊢ (∃x ∈ A (φ → ψ) ↔ (∀x ∈ A φ → ∃x ∈ A ψ))
4	2, 3	syl6rbbr 255	1 ⊢ (A ≠ ∅ → (∃x ∈ A (φ → ψ) ↔ (φ → ∃x ∈ A ψ)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ≠ wne 2517  ∀wral 2615  ∃wrex 2616  ∅c0 3551

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is used by:  ncssfin  6152