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Theorem nfbidf 1774
 Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.)
Hypotheses
Ref Expression
nfbidf.1 xφ
nfbidf.2 (φ → (ψχ))
Assertion
Ref Expression
nfbidf (φ → (Ⅎxψ ↔ Ⅎxχ))

Proof of Theorem nfbidf
StepHypRef Expression
1 nfbidf.1 . . 3 xφ
2 nfbidf.2 . . . 4 (φ → (ψχ))
31, 2albid 1772 . . . 4 (φ → (xψxχ))
42, 3imbi12d 311 . . 3 (φ → ((ψxψ) ↔ (χxχ)))
51, 4albid 1772 . 2 (φ → (x(ψxψ) ↔ x(χxχ)))
6 df-nf 1545 . 2 (Ⅎxψx(ψxψ))
7 df-nf 1545 . 2 (Ⅎxχx(χxχ))
85, 6, 73bitr4g 279 1 (φ → (Ⅎxψ ↔ Ⅎxχ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545 This theorem is referenced by:  nfsb4t  2080  dvelimdf  2082  nfcjust  2477  nfceqdf  2488
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