Theorem r19.2m 3546

Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1660). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) (Revised by Jim Kingdon, 7-Apr-2023.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem r19.2m

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2265	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
2	1	cbvexv 1941	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑧 𝑧 ∈ 𝐴)
3		eleq1w 2265	. . . 4 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
4	3	cbvexv 1941	. . 3 ⊢ (∃𝑧 𝑧 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ 𝐴)
5	2, 4	bitri 184	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ 𝐴)
6		df-ral 2488	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
7		exintr 1656	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
8	6, 7	sylbi 121	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
9		df-rex 2489	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
10	8, 9	imbitrrdi 162	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 𝜑))
11	10	impcom 125	. 2 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑)
12	5, 11	sylanbr 285	1 ⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1370  ∃wex 1514   ∈ wcel 2175  ∀wral 2483  ∃wrex 2484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556

This theorem depends on definitions:  df-bi 117  df-clel 2200  df-ral 2488  df-rex 2489

This theorem is referenced by:  intssunim  3906  riinm  3999  iinexgm  4197  xpiindim  4814  cnviinm  5223  eusvobj2  5929  iinerm  6693  suplocexprlemml  7828  rexfiuz  11242  r19.2uz  11246  climuni  11546  pc2dvds  12595  issubg4m  13471  cncnp2m  14645