Theorem drnf1 1969

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 1965	. . . 4 ⊢ (∀x x = y → (∀xφ ↔ ∀yψ))
3	1, 2	imbi12d 311	. . 3 ⊢ (∀x x = y → ((φ → ∀xφ) ↔ (ψ → ∀yψ)))
4	3	dral1 1965	. 2 ⊢ (∀x x = y → (∀x(φ → ∀xφ) ↔ ∀y(ψ → ∀yψ)))
5		df-nf 1545	. 2 ⊢ (Ⅎxφ ↔ ∀x(φ → ∀xφ))
6		df-nf 1545	. 2 ⊢ (Ⅎyψ ↔ ∀y(ψ → ∀yψ))
7	4, 5, 6	3bitr4g 279	1 ⊢ (∀x x = y → (Ⅎxφ ↔ Ⅎyψ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  nfald2  1972  drnfc1  2506