Theorem r19.2z 3640

Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1659). The restricted version is valid only when the domain of quantification is not empty. (Contributed by NM, 15-Nov-2003.)

Assertion

Ref	Expression
r19.2z	⊢ ((A ≠ ∅ ∧ ∀x ∈ A φ) → ∃x ∈ A φ)

Distinct variable group:   x,A

Allowed substitution hint:   φ(x)

Proof of Theorem r19.2z

Step	Hyp	Ref	Expression
1		df-ral 2620	. . . 4 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
2		exintr 1614	. . . 4 ⊢ (∀x(x ∈ A → φ) → (∃x x ∈ A → ∃x(x ∈ A ∧ φ)))
3	1, 2	sylbi 187	. . 3 ⊢ (∀x ∈ A φ → (∃x x ∈ A → ∃x(x ∈ A ∧ φ)))
4		n0 3560	. . 3 ⊢ (A ≠ ∅ ↔ ∃x x ∈ A)
5		df-rex 2621	. . 3 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
6	3, 4, 5	3imtr4g 261	. 2 ⊢ (∀x ∈ A φ → (A ≠ ∅ → ∃x ∈ A φ))
7	6	impcom 419	1 ⊢ ((A ≠ ∅ ∧ ∀x ∈ A φ) → ∃x ∈ A φ)

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   ∈ wcel 1710   ≠ wne 2517  ∀wral 2615  ∃wrex 2616  ∅c0 3551

This theorem was proved from 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 theorem 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 referenced by:  r19.2zb  3641  intssuni  3949  riinn0  4041