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Theorem eujustALT 2207
 Description: A soundness justification theorem for df-eu 2208, showing that the definition is equivalent to itself with its dummy variable renamed. Note that y and z needn't be distinct variables. While this isn't strictly necessary for soundness, the proof provides an example of how it can be achieved through the use of dvelim 2016. (Contributed by NM, 11-Mar-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
eujustALT (yx(φx = y) ↔ zx(φx = z))
Distinct variable groups:   x,y   x,z   φ,y   φ,z
Allowed substitution hint:   φ(x)

Proof of Theorem eujustALT
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 equequ2 1686 . . . . . 6 (y = z → (x = yx = z))
21bibi2d 309 . . . . 5 (y = z → ((φx = y) ↔ (φx = z)))
32albidv 1625 . . . 4 (y = z → (x(φx = y) ↔ x(φx = z)))
43sps 1754 . . 3 (y y = z → (x(φx = y) ↔ x(φx = z)))
54drex1 1967 . 2 (y y = z → (yx(φx = y) ↔ zx(φx = z)))
6 hbnae 1955 . . . . . 6 y y = zy ¬ y y = z)
7 hbnae 1955 . . . . . 6 y y = zz ¬ y y = z)
86, 7alrimih 1565 . . . . 5 y y = zyz ¬ y y = z)
9 ax-17 1616 . . . . . . . 8 x(φx = w) → z ¬ x(φx = w))
10 equequ2 1686 . . . . . . . . . . 11 (w = y → (x = wx = y))
1110bibi2d 309 . . . . . . . . . 10 (w = y → ((φx = w) ↔ (φx = y)))
1211albidv 1625 . . . . . . . . 9 (w = y → (x(φx = w) ↔ x(φx = y)))
1312notbid 285 . . . . . . . 8 (w = y → (¬ x(φx = w) ↔ ¬ x(φx = y)))
149, 13dvelim 2016 . . . . . . 7 z z = y → (¬ x(φx = y) → z ¬ x(φx = y)))
1514naecoms 1948 . . . . . 6 y y = z → (¬ x(φx = y) → z ¬ x(φx = y)))
16 ax-17 1616 . . . . . . 7 x(φx = w) → y ¬ x(φx = w))
17 equequ2 1686 . . . . . . . . . 10 (w = z → (x = wx = z))
1817bibi2d 309 . . . . . . . . 9 (w = z → ((φx = w) ↔ (φx = z)))
1918albidv 1625 . . . . . . . 8 (w = z → (x(φx = w) ↔ x(φx = z)))
2019notbid 285 . . . . . . 7 (w = z → (¬ x(φx = w) ↔ ¬ x(φx = z)))
2116, 20dvelim 2016 . . . . . 6 y y = z → (¬ x(φx = z) → y ¬ x(φx = z)))
223notbid 285 . . . . . . 7 (y = z → (¬ x(φx = y) ↔ ¬ x(φx = z)))
2322a1i 10 . . . . . 6 y y = z → (y = z → (¬ x(φx = y) ↔ ¬ x(φx = z))))
2415, 21, 23cbv2h 1980 . . . . 5 (yz ¬ y y = z → (y ¬ x(φx = y) ↔ z ¬ x(φx = z)))
258, 24syl 15 . . . 4 y y = z → (y ¬ x(φx = y) ↔ z ¬ x(φx = z)))
2625notbid 285 . . 3 y y = z → (¬ y ¬ x(φx = y) ↔ ¬ z ¬ x(φx = z)))
27 df-ex 1542 . . 3 (yx(φx = y) ↔ ¬ y ¬ x(φx = y))
28 df-ex 1542 . . 3 (zx(φx = z) ↔ ¬ z ¬ x(φx = z))
2926, 27, 283bitr4g 279 . 2 y y = z → (yx(φx = y) ↔ zx(φx = z)))
305, 29pm2.61i 156 1 (yx(φx = y) ↔ zx(φx = z))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545 This theorem is referenced by: (None)
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