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 ) ) |
| 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 ) ) |