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Theorem r19.2zb 3640
 Description: A response to the notion that the condition A ≠ ∅ can be removed in r19.2z 3639. Interestingly enough, φ does not figure in the left-hand side. (Contributed by Jeff Hankins, 24-Aug-2009.)
Assertion
Ref Expression
r19.2zb (A ↔ (x A φx A φ))
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem r19.2zb
StepHypRef Expression
1 r19.2z 3639 . . 3 ((A x A φ) → x A φ)
21ex 423 . 2 (A → (x A φx A φ))
3 noel 3554 . . . . . . 7 ¬ x
43pm2.21i 123 . . . . . 6 (x φ)
54rgen 2679 . . . . 5 x φ
6 raleq 2807 . . . . 5 (A = → (x A φx φ))
75, 6mpbiri 224 . . . 4 (A = x A φ)
87necon3bi 2557 . . 3 x A φA)
9 exsimpl 1592 . . . 4 (x(x A φ) → x x A)
10 df-rex 2620 . . . 4 (x A φx(x A φ))
11 n0 3559 . . . 4 (Ax x A)
129, 10, 113imtr4i 257 . . 3 (x A φA)
138, 12ja 153 . 2 ((x A φx A φ) → A)
142, 13impbii 180 1 (A ↔ (x A φx A φ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∀wral 2614  ∃wrex 2615  ∅c0 3550 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551 This theorem is referenced by: (None)
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