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Theorem rmoeq1f 2648
 Description: Equality theorem for restricted at-most-one quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Hypotheses
Ref Expression
raleq1f.1 𝑥𝐴
raleq1f.2 𝑥𝐵
Assertion
Ref Expression
rmoeq1f (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))

Proof of Theorem rmoeq1f
StepHypRef Expression
1 raleq1f.1 . . . 4 𝑥𝐴
2 raleq1f.2 . . . 4 𝑥𝐵
31, 2nfeq 2304 . . 3 𝑥 𝐴 = 𝐵
4 eleq2 2218 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
54anbi1d 461 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜑)))
63, 5mobid 2038 . 2 (𝐴 = 𝐵 → (∃*𝑥(𝑥𝐴𝜑) ↔ ∃*𝑥(𝑥𝐵𝜑)))
7 df-rmo 2440 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
8 df-rmo 2440 . 2 (∃*𝑥𝐵 𝜑 ↔ ∃*𝑥(𝑥𝐵𝜑))
96, 7, 83bitr4g 222 1 (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332  ∃*wmo 2004   ∈ wcel 2125  Ⅎwnfc 2283  ∃*wrmo 2435 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-cleq 2147  df-clel 2150  df-nfc 2285  df-rmo 2440 This theorem is referenced by:  rmoeq1  2652
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