Theorem drnf1 1721

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
drnf1	⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))

Proof of Theorem drnf1

Step	Hyp	Ref	Expression
1		drex2.1	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	dral1 1718	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
3	1, 2	imbi12d 233	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑦𝜓)))
4	3	dral1 1718	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜓 → ∀𝑦𝜓)))
5		df-nf 1449	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
6		df-nf 1449	. 2 ⊢ (Ⅎ𝑦𝜓 ↔ ∀𝑦(𝜓 → ∀𝑦𝜓))
7	4, 5, 6	3bitr4g 222	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1341  Ⅎwnf 1448

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by:  drnfc1  2325