Theorem rspcebdv 3591

Description: Restricted existential specialization, using implicit substitution in both directions. (Contributed by AV, 8-Jan-2022.)

Hypotheses

Ref	Expression
rspcdv.1	⊢ (𝜑 → 𝐴 ∈ 𝐵)
rspcdv.2	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
rspcebdv.1	⊢ ((𝜑 ∧ 𝜓) → 𝑥 = 𝐴)

Assertion

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

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

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspcebdv

Step	Hyp	Ref	Expression
1		rspcebdv.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑥 = 𝐴)
2		rspcdv.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
3	1, 2	syldan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (𝜓 ↔ 𝜒))
4	3	biimpd 229	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝜓 → 𝜒))
5	4	expcom 413	. . . 4 ⊢ (𝜓 → (𝜑 → (𝜓 → 𝜒)))
6	5	pm2.43b 55	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
7	6	rexlimdvw 3141	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 → 𝜒))
8		rspcdv.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
9	8, 2	rspcedv 3590	. 2 ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓))
10	7, 9	impbid 212	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3047  df-rex 3056

This theorem is referenced by:  fusgr2wsp2nb  30270