Theorem r19.27m 3511

Description: Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)

Hypothesis

Ref	Expression
r19.27m.1	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
r19.27m	⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem r19.27m

Step	Hyp	Ref	Expression
1		r19.26 2597	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))
2		r19.27m.1	. . . 4 ⊢ Ⅎ𝑥𝜓
3	2	r19.3rm 3504	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜓))
4	3	anbi2d 462	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))
5	1, 4	bitr4id 198	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  Ⅎwnf 1454  ∃wex 1486   ∈ wcel 2142  ∀wral 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-nf 1455  df-cleq 2164  df-clel 2167  df-ral 2454

This theorem is referenced by:  r19.27mv  3512  raaanlem  3521