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Metamath Proof Explorer

Theorem ceqsralt

Description: Restricted quantifier version of ceqsalt . (Contributed by NM, 28-Feb-2013) (Revised by Mario Carneiro, 10-Oct-2016)

Ref Expression
Assertion ceqsralt ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ 𝐴 ∈ 𝐵 ) → ( ∀ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 biimt ⊢ ( 𝐴 ∈ 𝐵 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ ( 𝐴 ∈ 𝐵 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) ) )
2 df-ral ⊢ ( ∀ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → ( 𝑥 = 𝐴 → 𝜑 ) ) )
3 eleq1 ⊢ ( 𝑥 = 𝐴 → ( 𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵 ) )
4 3 pm5.32ri ⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴 ) ↔ ( 𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴 ) )
5 4 imbi1i ⊢ ( ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴 ) → 𝜑 ) ↔ ( ( 𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴 ) → 𝜑 ) )
6 impexp ⊢ ( ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴 ) → 𝜑 ) ↔ ( 𝑥 ∈ 𝐵 → ( 𝑥 = 𝐴 → 𝜑 ) ) )
7 impexp ⊢ ( ( ( 𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴 ) → 𝜑 ) ↔ ( 𝐴 ∈ 𝐵 → ( 𝑥 = 𝐴 → 𝜑 ) ) )
8 5 6 7 3bitr3i ⊢ ( ( 𝑥 ∈ 𝐵 → ( 𝑥 = 𝐴 → 𝜑 ) ) ↔ ( 𝐴 ∈ 𝐵 → ( 𝑥 = 𝐴 → 𝜑 ) ) )
9 8 albii ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐵 → ( 𝑥 = 𝐴 → 𝜑 ) ) ↔ ∀ 𝑥 ( 𝐴 ∈ 𝐵 → ( 𝑥 = 𝐴 → 𝜑 ) ) )
10 19.21v ⊢ ( ∀ 𝑥 ( 𝐴 ∈ 𝐵 → ( 𝑥 = 𝐴 → 𝜑 ) ) ↔ ( 𝐴 ∈ 𝐵 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) )
11 2 9 10 3bitrri ⊢ ( ( 𝐴 ∈ 𝐵 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) ↔ ∀ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 → 𝜑 ) )
12 1 11 bitrdi ⊢ ( 𝐴 ∈ 𝐵 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 → 𝜑 ) ) )
13 12 3ad2ant3 ⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ 𝐴 ∈ 𝐵 ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 → 𝜑 ) ) )
14 ceqsalt ⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ 𝐴 ∈ 𝐵 ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) )
15 13 14 bitr3d ⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ 𝐴 ∈ 𝐵 ) → ( ∀ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) )