Theorem ralbinrald 43314

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 484	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝜓 ↔ 𝜃))
4	1, 3	rspcdv 3615	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → 𝜃))
5		ralbinrald.2	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → 𝑥 = 𝑋)
6	2	bicomd 225	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝜃 ↔ 𝜓))
7	5, 6	syl 17	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝜃 ↔ 𝜓))
8	7	adantl 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜃 ↔ 𝜓))
9	8	biimpd 231	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜃 → 𝜓))
10	9	ralrimdva 3189	. 2 ⊢ (𝜑 → (𝜃 → ∀𝑥 ∈ 𝐴 𝜓))
11	4, 10	impbid 214	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ 𝜃))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-cleq 2814  df-clel 2893  df-ral 3143

This theorem is referenced by:  dfdfat2  43320