Theorem dfrals2 17038

Description: The bounded "all some" form is the general form with the class membership folded into the antecedent. (Contributed by David A. Wheeler, 22-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)

Assertion

Ref	Expression
dfrals2	⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))

Proof of Theorem dfrals2

Step	Hyp	Ref	Expression
1		df-ral 2533	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
2		impexp 263	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
3	2	albii 1523	. . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
4	1, 3	bitr4i 187	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
5		df-rex 2534	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
6	4, 5	anbi12i 464	. 2 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃𝑥 ∈ 𝐴 𝜑) ↔ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ∧ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
7		df-rals 17037	. 2 ⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃𝑥 ∈ 𝐴 𝜑))
8		df-als 17036	. 2 ⊢ (∀∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ↔ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ∧ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
9	6, 7, 8	3bitr4i 212	1 ⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1400  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  ∀∃wals 17034  ∀∃wrals 17035

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 1500  ax-gen 1502

This theorem depends on definitions:  df-bi 117  df-ral 2533  df-rex 2534  df-als 17036  df-rals 17037

This theorem is referenced by:  rals-no-surprise  17056