Theorem ralbinrald 47747

Description: Elemination of a restricted universal quantification under certain conditions. (Contributed by Alexander van der Vekens, 2-Aug-2017.)

Hypotheses

Ref	Expression
ralbinrald.1	⊢ (𝜑 → 𝑋 ∈ 𝐴)
ralbinrald.2	⊢ (𝑥 ∈ 𝐴 → 𝑥 = 𝑋)
ralbinrald.3	⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜃))

Assertion

Ref	Expression
ralbinrald	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ 𝜃))

Distinct variable groups:   𝑥,𝑋   𝑥,𝐴   𝜑,𝑥   𝜃,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem ralbinrald

Step	Hyp	Ref	Expression
1		ralbinrald.1	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
2		ralbinrald.3	. . . 4 ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜃))
3	2	adantl 486	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝜓 ↔ 𝜃))
4	1, 3	rspcdv 3582	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → 𝜃))
5		ralbinrald.2	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → 𝑥 = 𝑋)
6	2	bicomd 226	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝜃 ↔ 𝜓))
7	5, 6	syl 18	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝜃 ↔ 𝜓))
8	7	adantl 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜃 ↔ 𝜓))
9	8	biimpd 232	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜃 → 𝜓))
10	9	ralrimdva 3171	. 2 ⊢ (𝜑 → (𝜃 → ∀𝑥 ∈ 𝐴 𝜓))
11	4, 10	impbid 215	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ 𝜃))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086

This theorem is referenced by:  dfdfat2  47753