Theorem rspcebdv 3565

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 594	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (𝜓 ↔ 𝜒))
4	3	biimpd 232	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝜓 → 𝜒))
5	4	expcom 417	. . . 4 ⊢ (𝜓 → (𝜑 → (𝜓 → 𝜒)))
6	5	pm2.43b 55	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
7	6	rexlimdvw 3249	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 → 𝜒))
8		rspcdv.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
9	8, 2	rspcedv 3564	. 2 ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓))
10	7, 9	impbid 215	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112

This theorem is referenced by:  fusgr2wsp2nb  28119