Theorem drnfc2OLD 2922

Description: Obsolete version of drnfc2 2921 as of 22-Sep-2024. (Contributed by Mario Carneiro, 8-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
drnfc1.1	⊢ (∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵)

Assertion

Ref	Expression
drnfc2OLD	⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝐴 ↔ Ⅎ𝑧𝐵))

Proof of Theorem drnfc2OLD

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		drnfc1.1	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵)
2	1	eleq2d 2815	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐵))
3	2	drnf2 2439	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧 𝑤 ∈ 𝐴 ↔ Ⅎ𝑧 𝑤 ∈ 𝐵))
4	3	albidv 1916	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑤Ⅎ𝑧 𝑤 ∈ 𝐴 ↔ ∀𝑤Ⅎ𝑧 𝑤 ∈ 𝐵))
5		df-nfc 2881	. 2 ⊢ (Ⅎ𝑧𝐴 ↔ ∀𝑤Ⅎ𝑧 𝑤 ∈ 𝐴)
6		df-nfc 2881	. 2 ⊢ (Ⅎ𝑧𝐵 ↔ ∀𝑤Ⅎ𝑧 𝑤 ∈ 𝐵)
7	4, 5, 6	3bitr4g 314	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝐴 ↔ Ⅎ𝑧𝐵))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1532   = wceq 1534  Ⅎwnf 1778   ∈ wcel 2099  Ⅎwnfc 2879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-13 2367  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-cleq 2720  df-clel 2806  df-nfc 2881

This theorem is referenced by: (None)