Theorem rspceb2dv 3626

Description: Restricted existential specialization, using implicit substitution in both directions. (Contributed by Zhi Wang, 28-Sep-2024.)

Hypotheses

Ref	Expression
rspceb2dv.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 → 𝜒))
rspceb2dv.2	⊢ ((𝜑 ∧ 𝜒) → 𝐴 ∈ 𝐵)
rspceb2dv.3	⊢ ((𝜑 ∧ 𝜒) → 𝜃)
rspceb2dv.4	⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))

Assertion

Ref	Expression
rspceb2dv	⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ 𝜒))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜒,𝑥   𝜑,𝑥   𝜃,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspceb2dv

Step	Hyp	Ref	Expression
1		rspceb2dv.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 → 𝜒))
2	1	rexlimdva 3153	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 → 𝜒))
3		rspceb2dv.2	. . . 4 ⊢ ((𝜑 ∧ 𝜒) → 𝐴 ∈ 𝐵)
4		rspceb2dv.3	. . . 4 ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
5		rspceb2dv.4	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))
6	5	rspcev 3622	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜃) → ∃𝑥 ∈ 𝐵 𝜓)
7	3, 4, 6	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝜒) → ∃𝑥 ∈ 𝐵 𝜓)
8	7	ex 412	. 2 ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓))
9	2, 8	impbid 212	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∃wrex 3068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069

This theorem is referenced by:  negfi  12215  psdmul  22188  uspgrlimlem1  47891  ipolubdm  48776  ipoglbdm  48779