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Theorem moxfr 38215
 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 2597 . . . . . . . 8 (∃!𝑦 𝑥 = 𝐴 → ∃𝑦 𝑥 = 𝐴)
53, 4ax-mp 5 . . . . . . 7 𝑦 𝑥 = 𝐴
6 rexv 3422 . . . . . . 7 (∃𝑦 ∈ V 𝑥 = 𝐴 ↔ ∃𝑦 𝑥 = 𝐴)
75, 6mpbir 223 . . . . . 6 𝑦 ∈ V 𝑥 = 𝐴
87a1i 11 . . . . 5 (𝑥 ∈ V → ∃𝑦 ∈ V 𝑥 = 𝐴)
9 moxfr.c . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
102, 8, 9rexxfr 5128 . . . 4 (∃𝑥 ∈ V 𝜑 ↔ ∃𝑦 ∈ V 𝜓)
11 rexv 3422 . . . 4 (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
12 rexv 3422 . . . 4 (∃𝑦 ∈ V 𝜓 ↔ ∃𝑦𝜓)
1310, 11, 123bitr3i 293 . . 3 (∃𝑥𝜑 ↔ ∃𝑦𝜓)
141, 3, 9euxfr 3604 . . 3 (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
1513, 14imbi12i 342 . 2 ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
16 moeu 2603 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
17 moeu 2603 . 2 (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
1815, 16, 173bitr4i 295 1 (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   = wceq 1601  ∃wex 1823   ∈ wcel 2107  ∃*wmo 2549  ∃!weu 2586  ∃wrex 3091  Vcvv 3398 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-v 3400 This theorem is referenced by: (None)
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