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Theorem cbvrexcsf 3199
 Description: A more general version of cbvrexf 2830 that has no distinct variable restrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario Carneiro, 7-Dec-2014.)
Hypotheses
Ref Expression
cbvralcsf.1 yA
cbvralcsf.2 xB
cbvralcsf.3 yφ
cbvralcsf.4 xψ
cbvralcsf.5 (x = yA = B)
cbvralcsf.6 (x = y → (φψ))
Assertion
Ref Expression
cbvrexcsf (x A φy B ψ)

Proof of Theorem cbvrexcsf
StepHypRef Expression
1 cbvralcsf.1 . . . 4 yA
2 cbvralcsf.2 . . . 4 xB
3 cbvralcsf.3 . . . . 5 yφ
43nfn 1793 . . . 4 y ¬ φ
5 cbvralcsf.4 . . . . 5 xψ
65nfn 1793 . . . 4 x ¬ ψ
7 cbvralcsf.5 . . . 4 (x = yA = B)
8 cbvralcsf.6 . . . . 5 (x = y → (φψ))
98notbid 285 . . . 4 (x = y → (¬ φ ↔ ¬ ψ))
101, 2, 4, 6, 7, 9cbvralcsf 3198 . . 3 (x A ¬ φy B ¬ ψ)
1110notbii 287 . 2 x A ¬ φ ↔ ¬ y B ¬ ψ)
12 dfrex2 2627 . 2 (x A φ ↔ ¬ x A ¬ φ)
13 dfrex2 2627 . 2 (y B ψ ↔ ¬ y B ¬ ψ)
1411, 12, 133bitr4i 268 1 (x A φy B 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-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-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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-sbc 3047  df-csb 3137 This theorem is referenced by:  cbvrexv2  3203
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