Theorem ralbidar 44464

Description: More general form of ralbida 3270. (Contributed by Andrew Salmon, 25-Jul-2011.)

Hypotheses

Ref	Expression
ralbidar.1	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜑)
ralbidar.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
ralbidar	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))

Proof of Theorem ralbidar

Step	Hyp	Ref	Expression
1		ralbidar.1	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜑)
2		ralbidar.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
3	2	ex 412	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒)))
4	3	ralimi 3083	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒)))
5	1, 4	syl 17	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒)))
6		df-ral 3062	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒))))
7	5, 6	sylib 218	. . 3 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒))))
8		pm2.43 56	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒))) → (𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒)))
9	8	pm5.74d 273	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒))) → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐴 → 𝜒)))
10	9	alimi 1811	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒))) → ∀𝑥((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐴 → 𝜒)))
11		albi 1818	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐴 → 𝜒)) → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜒)))
12	7, 10, 11	3syl 18	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜒)))
13		df-ral 3062	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
14		df-ral 3062	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜒))
15	12, 13, 14	3bitr4g 314	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   ∈ wcel 2108  ∀wral 3061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-an 396  df-ral 3062

This theorem is referenced by: (None)