Metamath Proof Explorer

Theorem drnfc1OLD

Description: Obsolete version of drnfc1 as of 22-Sep-2024. (Contributed by Mario Carneiro, 8-Oct-2016) Avoid ax-11 . (Revised by Wolf Lammen, 10-May-2023) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Hypothesis drnfc1.1 ⊢ ∀ x x = y → A = B
Assertion drnfc1OLD ⊢ ∀ x x = y → Ⅎ _ x A ↔ Ⅎ _ y B

Proof

Step Hyp Ref Expression
1 drnfc1.1 ⊢ ∀ x x = y → A = B
2 1 eleq2d ⊢ ∀ x x = y → w ∈ A ↔ w ∈ B
3 2 drnf1 ⊢ ∀ x x = y → Ⅎ x w ∈ A ↔ Ⅎ y w ∈ B
4 3 albidv ⊢ ∀ x x = y → ∀ w Ⅎ x w ∈ A ↔ ∀ w Ⅎ y w ∈ B
5 df-nfc ⊢ Ⅎ _ x A ↔ ∀ w Ⅎ x w ∈ A
6 df-nfc ⊢ Ⅎ _ y B ↔ ∀ w Ⅎ y w ∈ B
7 4 5 6 3bitr4g ⊢ ∀ x x = y → Ⅎ _ x A ↔ Ⅎ _ y B