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Theorem 19.43 1501
Description: Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)
Assertion
Ref Expression
19.43 (x(φ ψ) ↔ (xφ xψ))

Proof of Theorem 19.43
StepHypRef Expression
1 hbe1 1365 . . . 4 (xφxxφ)
2 hbe1 1365 . . . 4 (xψxxψ)
31, 2hbor 1420 . . 3 ((xφ xψ) → x(xφ xψ))
4 19.8a 1464 . . . 4 (φxφ)
5 19.8a 1464 . . . 4 (ψxψ)
64, 5orim12i 663 . . 3 ((φ ψ) → (xφ xψ))
73, 6exlimih 1466 . 2 (x(φ ψ) → (xφ xψ))
8 orc 620 . . . 4 (φ → (φ ψ))
98eximi 1473 . . 3 (xφx(φ ψ))
10 olc 619 . . . 4 (ψ → (φ ψ))
1110eximi 1473 . . 3 (xψx(φ ψ))
129, 11jaoi 623 . 2 ((xφ xψ) → x(φ ψ))
137, 12impbii 117 1 (x(φ ψ) ↔ (xφ xψ))
Colors of variables: wff set class
Syntax hints:  wb 98   wo 616  wex 1362
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 617  ax-5 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-ial 1409
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  19.44  1554  19.45  1555  19.34  1556  sborv  1752  r19.43  2446  rexun  3100  unipr  3568  uniun  3573  unopab  3810  dmun  4469  coundi  4749  coundir  4750
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