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Theorem exdistrfor 1664
 Description: Distribution of existential quantifiers, with a bound-variable hypothesis saying that y is not free in φ, but x can be free in φ (and there is no distinct variable condition on x and y). (Contributed by Jim Kingdon, 25-Feb-2018.)
Hypothesis
Ref Expression
exdistrfor.1 (x x = y xyφ)
Assertion
Ref Expression
exdistrfor (xy(φ ψ) → x(φ yψ))

Proof of Theorem exdistrfor
StepHypRef Expression
1 exdistrfor.1 . 2 (x x = y xyφ)
2 biidd 161 . . . . . 6 (x x = y → ((φ ψ) ↔ (φ ψ)))
32drex1 1662 . . . . 5 (x x = y → (x(φ ψ) ↔ y(φ ψ)))
43drex2 1603 . . . 4 (x x = y → (xx(φ ψ) ↔ xy(φ ψ)))
5 hbe1 1366 . . . . . 6 (x(φ ψ) → xx(φ ψ))
6519.9h 1518 . . . . 5 (xx(φ ψ) ↔ x(φ ψ))
7 19.8a 1466 . . . . . . 7 (ψyψ)
87anim2i 324 . . . . . 6 ((φ ψ) → (φ yψ))
98eximi 1475 . . . . 5 (x(φ ψ) → x(φ yψ))
106, 9sylbi 114 . . . 4 (xx(φ ψ) → x(φ yψ))
114, 10syl6bir 153 . . 3 (x x = y → (xy(φ ψ) → x(φ yψ)))
12 ax-ial 1410 . . . 4 (xyφxxyφ)
13 19.40 1506 . . . . . 6 (y(φ ψ) → (yφ yψ))
14 19.9t 1517 . . . . . . . 8 (Ⅎyφ → (yφφ))
1514biimpd 132 . . . . . . 7 (Ⅎyφ → (yφφ))
1615anim1d 319 . . . . . 6 (Ⅎyφ → ((yφ yψ) → (φ yψ)))
1713, 16syl5 28 . . . . 5 (Ⅎyφ → (y(φ ψ) → (φ yψ)))
1817sps 1413 . . . 4 (xyφ → (y(φ ψ) → (φ yψ)))
1912, 18eximdh 1486 . . 3 (xyφ → (xy(φ ψ) → x(φ yψ)))
2011, 19jaoi 623 . 2 ((x x = y xyφ) → (xy(φ ψ) → x(φ yψ)))
211, 20ax-mp 7 1 (xy(φ ψ) → x(φ yψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∨ wo 616  ∀wal 1226  Ⅎwnf 1329  ∃wex 1363 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-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410 This theorem depends on definitions:  df-bi 110  df-nf 1330 This theorem is referenced by:  oprabidlem  5458
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