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Theorem ax4g 149
 Description: If F is true for all x:α, then it is true for A. (Contributed by Mario Carneiro, 9-Oct-2014.)
Hypotheses
Ref Expression
ax4g.1 F:(α → ∗)
ax4g.2 A:α
Assertion
Ref Expression
ax4g (F)⊧(FA)

Proof of Theorem ax4g
Dummy variable p is distinct from all other variables.
StepHypRef Expression
1 wal 134 . . . 4 :((α → ∗) → ∗)
2 ax4g.1 . . . 4 F:(α → ∗)
31, 2wc 50 . . 3 (F):∗
43trud 27 . 2 (F)⊧⊤
5 ax4g.2 . . . 4 A:α
62, 5wc 50 . . 3 (FA):∗
74ax-cb1 29 . . . . . 6 (F):∗
87id 25 . . . . 5 (F)⊧(F)
92alval 142 . . . . . 6 ⊤⊧[(F) = [F = λp:α ⊤]]
107, 9a1i 28 . . . . 5 (F)⊧[(F) = [F = λp:α ⊤]]
118, 10mpbi 82 . . . 4 (F)⊧[F = λp:α ⊤]
122, 5, 11ceq1 89 . . 3 (F)⊧[(FA) = (λp:αA)]
135, 4hbth 109 . . 3 (F)⊧[(λp:αA) = ⊤]
146, 12, 13eqtri 95 . 2 (F)⊧[(FA) = ⊤]
154, 14mpbir 87 1 (F)⊧(FA)
 Colors of variables: type var term Syntax hints:   → ht 2  ∗hb 3  kc 5  λkl 6   = ke 7  ⊤kt 8  [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:  ax4  150  cla4v  152
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