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Theorem sbequilem 1716
Description: Propositional logic lemma used in the sbequi 1717 proof. (Contributed by Jim Kingdon, 1-Feb-2018.)
Hypotheses
Ref Expression
sbequilem.1 (φ (ψ → (χθ)))
sbequilem.2 (τ (ψ → (θη)))
Assertion
Ref Expression
sbequilem (φ (τ (ψ → (χη))))

Proof of Theorem sbequilem
StepHypRef Expression
1 sbequilem.1 . . . . . . . . . 10 (φ (ψ → (χθ)))
2 sbequilem.2 . . . . . . . . . 10 (τ (ψ → (θη)))
31, 2pm3.2i 257 . . . . . . . . 9 ((φ (ψ → (χθ))) (τ (ψ → (θη))))
4 andi 730 . . . . . . . . 9 (((φ (ψ → (χθ))) (τ (ψ → (θη)))) ↔ (((φ (ψ → (χθ))) τ) ((φ (ψ → (χθ))) (ψ → (θη)))))
53, 4mpbi 133 . . . . . . . 8 (((φ (ψ → (χθ))) τ) ((φ (ψ → (χθ))) (ψ → (θη))))
6 andir 731 . . . . . . . . 9 (((φ (ψ → (χθ))) τ) ↔ ((φ τ) ((ψ → (χθ)) τ)))
7 andir 731 . . . . . . . . 9 (((φ (ψ → (χθ))) (ψ → (θη))) ↔ ((φ (ψ → (θη))) ((ψ → (χθ)) (ψ → (θη)))))
86, 7orbi12i 680 . . . . . . . 8 ((((φ (ψ → (χθ))) τ) ((φ (ψ → (χθ))) (ψ → (θη)))) ↔ (((φ τ) ((ψ → (χθ)) τ)) ((φ (ψ → (θη))) ((ψ → (χθ)) (ψ → (θη))))))
95, 8mpbi 133 . . . . . . 7 (((φ τ) ((ψ → (χθ)) τ)) ((φ (ψ → (θη))) ((ψ → (χθ)) (ψ → (θη)))))
10 pm3.43 534 . . . . . . . . . 10 (((ψ → (χθ)) (ψ → (θη))) → (ψ → ((χθ) (θη))))
11 pm3.33 327 . . . . . . . . . 10 (((χθ) (θη)) → (χη))
1210, 11syl6 29 . . . . . . . . 9 (((ψ → (χθ)) (ψ → (θη))) → (ψ → (χη)))
1312orim2i 677 . . . . . . . 8 (((φ (ψ → (θη))) ((ψ → (χθ)) (ψ → (θη)))) → ((φ (ψ → (θη))) (ψ → (χη))))
1413orim2i 677 . . . . . . 7 ((((φ τ) ((ψ → (χθ)) τ)) ((φ (ψ → (θη))) ((ψ → (χθ)) (ψ → (θη))))) → (((φ τ) ((ψ → (χθ)) τ)) ((φ (ψ → (θη))) (ψ → (χη)))))
159, 14ax-mp 7 . . . . . 6 (((φ τ) ((ψ → (χθ)) τ)) ((φ (ψ → (θη))) (ψ → (χη))))
16 simpr 103 . . . . . . . 8 (((φ (ψ → (χθ))) τ) → τ)
176, 16sylbir 125 . . . . . . 7 (((φ τ) ((ψ → (χθ)) τ)) → τ)
1817orim1i 676 . . . . . 6 ((((φ τ) ((ψ → (χθ)) τ)) ((φ (ψ → (θη))) (ψ → (χη)))) → (τ ((φ (ψ → (θη))) (ψ → (χη)))))
1915, 18ax-mp 7 . . . . 5 (τ ((φ (ψ → (θη))) (ψ → (χη))))
20 simpl 102 . . . . . . 7 ((φ (ψ → (θη))) → φ)
2120orim1i 676 . . . . . 6 (((φ (ψ → (θη))) (ψ → (χη))) → (φ (ψ → (χη))))
2221orim2i 677 . . . . 5 ((τ ((φ (ψ → (θη))) (ψ → (χη)))) → (τ (φ (ψ → (χη)))))
2319, 22ax-mp 7 . . . 4 (τ (φ (ψ → (χη))))
24 orass 683 . . . 4 (((τ φ) (ψ → (χη))) ↔ (τ (φ (ψ → (χη)))))
2523, 24mpbir 134 . . 3 ((τ φ) (ψ → (χη)))
26 orcom 646 . . . 4 ((τ φ) ↔ (φ τ))
2726orbi1i 679 . . 3 (((τ φ) (ψ → (χη))) ↔ ((φ τ) (ψ → (χη))))
2825, 27mpbi 133 . 2 ((φ τ) (ψ → (χη)))
29 orass 683 . 2 (((φ τ) (ψ → (χη))) ↔ (φ (τ (ψ → (χη)))))
3028, 29mpbi 133 1 (φ (τ (ψ → (χη))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wo 628
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  sbequi  1717
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