Theorem r19.2z 1766

Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 713). The restricted version is valid only when the domain of quantification is not empty.

Assertion

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

Distinct variable group(s):   x,A

Proof of Theorem r19.2z

Step	Hyp	Ref	Expression
1		df-ral 1205	. . . 4 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
2		exintr 793	. . . 4 ⊢ (∀x(x ∈ A → φ) → (∃x x ∈ A → ∃x(x ∈ A ∧ φ)))
3	1, 2	sylbi 174	. . 3 ⊢ (∀x ∈ A φ → (∃x x ∈ A → ∃x(x ∈ A ∧ φ)))
4		n0 1714	. . 3 ⊢ (¬ A = ∅ ↔ ∃x x ∈ A)
5		df-rex 1206	. . 3 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
6	3, 4, 5	3imtr4g 426	. 2 ⊢ (∀x ∈ A φ → (¬ A = ∅ → ∃x ∈ A φ))
7	6	com12 13	1 ⊢ (¬ A = ∅ → (∀x ∈ A φ → ∃x ∈ A φ))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  ∅c0 1707

This theorem is referenced by:  cfsuc 3709

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-nul 1708

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