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Theorem cla4v 152
 Description: If A(x) is true for all x:α, then it is true for C = A(B). (Contributed by Mario Carneiro, 9-Oct-2014.)
Hypotheses
Ref Expression
cla4v.1 A:∗
cla4v.2 B:α
cla4v.3 [x:α = B]⊧[A = C]
Assertion
Ref Expression
cla4v (λx:α A)⊧C
Distinct variable groups:   x,B   x,C   α,x

Proof of Theorem cla4v
StepHypRef Expression
1 cla4v.1 . . . 4 A:∗
21wl 66 . . 3 λx:α A:(α → ∗)
3 cla4v.2 . . 3 B:α
42, 3ax4g 149 . 2 (λx:α A)⊧(λx:α AB)
54ax-cb1 29 . . 3 (λx:α A):∗
6 cla4v.3 . . . 4 [x:α = B]⊧[A = C]
71, 3, 6cl 116 . . 3 ⊤⊧[(λx:α AB) = C]
85, 7a1i 28 . 2 (λx:α A)⊧[(λx:α AB) = C]
94, 8mpbi 82 1 (λx:α A)⊧C
 Colors of variables: type var term Syntax hints:  tv 1  ∗hb 3  kc 5  λkl 6   = ke 7  [kbr 9  ⊧wffMMJ2 11  wffMMJ2t 12  ∀tal 122 This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-simpl 20  ax-simpr 21  ax-id 24  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-wctl 31  ax-wctr 32  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-ded 46  ax-wct 47  ax-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113 This theorem depends on definitions:  df-ov 73  df-al 126 This theorem is referenced by:  pm2.21  153  ecase  163  exlimdv2  166  ax4e  168  eta  178  exlimd  183  ac  197  ax10  213
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