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Theorem cbvrmo 2526
Description: Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.)
Hypotheses
Ref Expression
cbvral.1 yφ
cbvral.2 xψ
cbvral.3 (x = y → (φψ))
Assertion
Ref Expression
cbvrmo (∃*x A φ∃*y A ψ)
Distinct variable groups:   x,A   y,A
Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem cbvrmo
StepHypRef Expression
1 cbvral.1 . . . 4 yφ
2 cbvral.2 . . . 4 xψ
3 cbvral.3 . . . 4 (x = y → (φψ))
41, 2, 3cbvrex 2524 . . 3 (x A φy A ψ)
51, 2, 3cbvreu 2525 . . 3 (∃!x A φ∃!y A ψ)
64, 5imbi12i 228 . 2 ((x A φ∃!x A φ) ↔ (y A ψ∃!y A ψ))
7 rmo5 2519 . 2 (∃*x A φ ↔ (x A φ∃!x A φ))
8 rmo5 2519 . 2 (∃*y A ψ ↔ (y A ψ∃!y A ψ))
96, 7, 83bitr4i 201 1 (∃*x A φ∃*y A ψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wnf 1346  wrex 2301  ∃!wreu 2302  ∃*wrmo 2303
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-reu 2307  df-rmo 2308
This theorem is referenced by:  cbvrmov  2530  cbvdisj  3746
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