Theorem drnf2 1712

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)

Hypothesis

Ref	Expression
drex2.1	⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
drnf2	⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓))

Proof of Theorem drnf2

Step	Hyp	Ref	Expression
1		drex2.1	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	dral2 1709	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))
3	1, 2	imbi12d 233	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝜑 → ∀𝑧𝜑) ↔ (𝜓 → ∀𝑧𝜓)))
4	3	dral2 1709	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑧(𝜑 → ∀𝑧𝜑) ↔ ∀𝑧(𝜓 → ∀𝑧𝜓)))
5		df-nf 1437	. 2 ⊢ (Ⅎ𝑧𝜑 ↔ ∀𝑧(𝜑 → ∀𝑧𝜑))
6		df-nf 1437	. 2 ⊢ (Ⅎ𝑧𝜓 ↔ ∀𝑧(𝜓 → ∀𝑧𝜓))
7	4, 5, 6	3bitr4g 222	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1329  Ⅎwnf 1436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-nf 1437

This theorem is referenced by:  nfsbxy  1913  nfsbxyt  1914  drnfc2  2297