Theorem r19.2m 3309

Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1529). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem r19.2m

Step	Hyp	Ref	Expression
1		df-ral 2311	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
2		exintr 1525	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
3	1, 2	sylbi 114	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
4		df-rex 2312	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
5	3, 4	syl6ibr 151	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 𝜑))
6	5	impcom 116	1 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241  ∃wex 1381   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-ral 2311  df-rex 2312

This theorem is referenced by:  intssunim  3637  riinm  3729  trintssm  3870  iinexgm  3908  xpiindim  4473  cnviinm  4859  eusvobj2  5498  iinerm  6178  r19.2uz  9591  climuni  9814