Theorem drnfc2

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Proof revision is marked as discouraged because the minimizer replaces albidv with dral2 , leading to a one byte longer proof. However feel free to manually edit it according to conventions. (TODO: dral2 depends on ax-13 , hence its usage during minimizing is discouraged. Check in the long run whether this is a permanent restriction). Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Mario Carneiro, 8-Oct-2016) Avoid ax-8 . (Revised by Wolf Lammen, 22-Sep-2024) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	drnfc1.1	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵 )
Assertion	drnfc2	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( Ⅎ 𝑧 𝐴 ↔ Ⅎ 𝑧 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		drnfc1.1	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵 )
2		eleq2w2	⊢ ( 𝐴 = 𝐵 → ( 𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐵 ) )
3	1 2	syl	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐵 ) )
4	3	drnf2	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( Ⅎ 𝑧 𝑤 ∈ 𝐴 ↔ Ⅎ 𝑧 𝑤 ∈ 𝐵 ) )
5	4	albidv	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑤 Ⅎ 𝑧 𝑤 ∈ 𝐴 ↔ ∀ 𝑤 Ⅎ 𝑧 𝑤 ∈ 𝐵 ) )
6		df-nfc	⊢ ( Ⅎ 𝑧 𝐴 ↔ ∀ 𝑤 Ⅎ 𝑧 𝑤 ∈ 𝐴 )
7		df-nfc	⊢ ( Ⅎ 𝑧 𝐵 ↔ ∀ 𝑤 Ⅎ 𝑧 𝑤 ∈ 𝐵 )
8	5 6 7	3bitr4g	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( Ⅎ 𝑧 𝐴 ↔ Ⅎ 𝑧 𝐵 ) )