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Theorem eqcomx 52
 Description: Commutativity of equality. (Contributed by Mario Carneiro, 7-Oct-2014.)
Hypotheses
Ref Expression
eqcomx.1 A:α
eqcomx.2 B:α
eqcomx.3 R⊧(( = A)B)
Assertion
Ref Expression
eqcomx R⊧(( = B)A)

Proof of Theorem eqcomx
StepHypRef Expression
1 eqcomx.3 . . . 4 R⊧(( = A)B)
21ax-cb1 29 . . 3 R:∗
3 eqcomx.1 . . . 4 A:α
43ax-refl 42 . . 3 ⊤⊧(( = A)A)
52, 4a1i 28 . 2 R⊧(( = A)A)
6 weq 41 . . . . . 6 = :(α → (α → ∗))
76ax-refl 42 . . . . 5 ⊤⊧(( = = ) = )
82, 7a1i 28 . . . 4 R⊧(( = = ) = )
9 eqcomx.2 . . . . 5 B:α
106, 6, 3, 9ax-ceq 51 . . . 4 ((( = = ) = ), (( = A)B))⊧(( = ( = A))( = B))
118, 1, 10syl2anc 19 . . 3 R⊧(( = ( = A))( = B))
126, 3wc 50 . . . 4 ( = A):(α → ∗)
136, 9wc 50 . . . 4 ( = B):(α → ∗)
1412, 13, 3, 3ax-ceq 51 . . 3 ((( = ( = A))( = B)), (( = A)A))⊧(( = (( = A)A))(( = B)A))
1511, 5, 14syl2anc 19 . 2 R⊧(( = (( = A)A))(( = B)A))
165, 15ax-eqmp 45 1 R⊧(( = B)A)
 Colors of variables: type var term Syntax hints:   → ht 2  ∗hb 3  kc 5   = ke 7  ⊧wffMMJ2 11  wffMMJ2t 12 This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-trud 26  ax-cb1 29  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-wc 49  ax-ceq 51 This theorem is referenced by:  mpbirx  53  eqcomi  79
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