Theorem drnf1 2420

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

Hypothesis

Ref	Expression
dral1.1	⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
drnf1	⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))

Proof of Theorem drnf1

Step	Hyp	Ref	Expression
1		dral1.1	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	dral1 2416	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
3	1, 2	imbi12d 346	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑦𝜓)))
4	3	dral1 2416	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜓 → ∀𝑦𝜓)))
5		nf5 2254	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
6		nf5 2254	. 2 ⊢ (Ⅎ𝑦𝜓 ↔ ∀𝑦(𝜓 → ∀𝑦𝜓))
7	4, 5, 6	3bitr4g 315	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1518  Ⅎwnf 1763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-10 2110  ax-12 2139  ax-13 2342

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1760  df-nf 1764

This theorem is referenced by:  nfald2  2422  drnfc1  2964  drnfc1OLD  2965  wl-nfs1t  34255