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Theorem cbvrexf 2830
 Description: Rule used to change bound variables, using implicit substitution. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.)
Hypotheses
Ref Expression
cbvralf.1 xA
cbvralf.2 yA
cbvralf.3 yφ
cbvralf.4 xψ
cbvralf.5 (x = y → (φψ))
Assertion
Ref Expression
cbvrexf (x A φy A ψ)

Proof of Theorem cbvrexf
StepHypRef Expression
1 cbvralf.1 . . . 4 xA
2 cbvralf.2 . . . 4 yA
3 cbvralf.3 . . . . 5 yφ
43nfn 1793 . . . 4 y ¬ φ
5 cbvralf.4 . . . . 5 xψ
65nfn 1793 . . . 4 x ¬ ψ
7 cbvralf.5 . . . . 5 (x = y → (φψ))
87notbid 285 . . . 4 (x = y → (¬ φ ↔ ¬ ψ))
91, 2, 4, 6, 8cbvralf 2829 . . 3 (x A ¬ φy A ¬ ψ)
109notbii 287 . 2 x A ¬ φ ↔ ¬ y A ¬ ψ)
11 dfrex2 2627 . 2 (x A φ ↔ ¬ x A ¬ φ)
12 dfrex2 2627 . 2 (y A ψ ↔ ¬ y A ¬ ψ)
1310, 11, 123bitr4i 268 1 (x A φy A ψ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  Ⅎwnf 1544   = wceq 1642  Ⅎwnfc 2476  ∀wral 2614  ∃wrex 2615 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  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620 This theorem is referenced by:  cbvrex  2832
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