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Theorem exdistrfor 1678
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 1676 . . . . 5 (x x = y → (x(φ ψ) ↔ y(φ ψ)))
43drex2 1617 . . . 4 (x x = y → (xx(φ ψ) ↔ xy(φ ψ)))
5 hbe1 1381 . . . . . 6 (x(φ ψ) → xx(φ ψ))
6519.9h 1531 . . . . 5 (xx(φ ψ) ↔ x(φ ψ))
7 19.8a 1479 . . . . . . 7 (ψyψ)
87anim2i 324 . . . . . 6 ((φ ψ) → (φ yψ))
98eximi 1488 . . . . 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 1424 . . . 4 (xyφxxyφ)
13 19.40 1519 . . . . . 6 (y(φ ψ) → (yφ yψ))
14 19.9t 1530 . . . . . . . 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 1427 . . . 4 (xyφ → (y(φ ψ) → (φ yψ)))
1912, 18eximdh 1499 . . 3 (xyφ → (xy(φ ψ) → x(φ yψ)))
2011, 19jaoi 635 . 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 628  wal 1240  wnf 1346  wex 1378
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  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-nf 1347
This theorem is referenced by:  oprabidlem  5476
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