Theorem drnf2 1748

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 1745	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))
3	1, 2	imbi12d 234	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝜑 → ∀𝑧𝜑) ↔ (𝜓 → ∀𝑧𝜓)))
4	3	dral2 1745	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑧(𝜑 → ∀𝑧𝜑) ↔ ∀𝑧(𝜓 → ∀𝑧𝜓)))
5		df-nf 1475	. 2 ⊢ (Ⅎ𝑧𝜑 ↔ ∀𝑧(𝜑 → ∀𝑧𝜑))
6		df-nf 1475	. 2 ⊢ (Ⅎ𝑧𝜓 ↔ ∀𝑧(𝜓 → ∀𝑧𝜓))
7	4, 5, 6	3bitr4g 223	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362  Ⅎwnf 1474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-nf 1475

This theorem is referenced by:  nfsbxy  1961  nfsbxyt  1962  drnfc2  2357