Theorem drnf2 1970

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
drnf2	⊢ (∀x x = y → (Ⅎzφ ↔ Ⅎzψ))

Proof of Theorem drnf2

Step	Hyp	Ref	Expression
1		dral1.1	. . . 4 ⊢ (∀x x = y → (φ ↔ ψ))
2	1	dral2 1966	. . . 4 ⊢ (∀x x = y → (∀zφ ↔ ∀zψ))
3	1, 2	imbi12d 311	. . 3 ⊢ (∀x x = y → ((φ → ∀zφ) ↔ (ψ → ∀zψ)))
4	3	dral2 1966	. 2 ⊢ (∀x x = y → (∀z(φ → ∀zφ) ↔ ∀z(ψ → ∀zψ)))
5		df-nf 1545	. 2 ⊢ (Ⅎzφ ↔ ∀z(φ → ∀zφ))
6		df-nf 1545	. 2 ⊢ (Ⅎzψ ↔ ∀z(ψ → ∀zψ))
7	4, 5, 6	3bitr4g 279	1 ⊢ (∀x x = y → (Ⅎzφ ↔ Ⅎzψ))

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:  nfsb4t  2080  drnfc2  2507