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Theorem moxfr 37074
Description: Transfer at-most-one between related expressions. (Contributed by Stefan O'Rear, 12-Feb-2015.)
Hypotheses
Ref Expression
moxfr.a 𝐴 ∈ V
moxfr.b ∃!𝑦 𝑥 = 𝐴
moxfr.c (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
moxfr (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
Distinct variable groups:   𝜓,𝑥   𝜑,𝑦   𝑥,𝐴   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑦)

Proof of Theorem moxfr
StepHypRef Expression
1 moxfr.a . . . . . 6 𝐴 ∈ V
21a1i 11 . . . . 5 (𝑦 ∈ V → 𝐴 ∈ V)
3 moxfr.b . . . . . . . 8 ∃!𝑦 𝑥 = 𝐴
4 euex 2492 . . . . . . . 8 (∃!𝑦 𝑥 = 𝐴 → ∃𝑦 𝑥 = 𝐴)
53, 4ax-mp 5 . . . . . . 7 𝑦 𝑥 = 𝐴
6 rexv 3215 . . . . . . 7 (∃𝑦 ∈ V 𝑥 = 𝐴 ↔ ∃𝑦 𝑥 = 𝐴)
75, 6mpbir 221 . . . . . 6 𝑦 ∈ V 𝑥 = 𝐴
87a1i 11 . . . . 5 (𝑥 ∈ V → ∃𝑦 ∈ V 𝑥 = 𝐴)
9 moxfr.c . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
102, 8, 9rexxfr 4879 . . . 4 (∃𝑥 ∈ V 𝜑 ↔ ∃𝑦 ∈ V 𝜓)
11 rexv 3215 . . . 4 (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
12 rexv 3215 . . . 4 (∃𝑦 ∈ V 𝜓 ↔ ∃𝑦𝜓)
1310, 11, 123bitr3i 290 . . 3 (∃𝑥𝜑 ↔ ∃𝑦𝜓)
141, 3, 9euxfr 3386 . . 3 (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
1513, 14imbi12i 340 . 2 ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
16 df-mo 2473 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
17 df-mo 2473 . 2 (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
1815, 16, 173bitr4i 292 1 (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1481  wex 1702  wcel 1988  ∃!weu 2468  ∃*wmo 2469  wrex 2910  Vcvv 3195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-v 3197
This theorem is referenced by: (None)
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