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Theorem cbvrmo 2834
 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 2832 . . 3 (x A φy A ψ)
51, 2, 3cbvreu 2833 . . 3 (∃!x A φ∃!y A ψ)
64, 5imbi12i 316 . 2 ((x A φ∃!x A φ) ↔ (y A ψ∃!y A ψ))
7 rmo5 2827 . 2 (∃*x A φ ↔ (x A φ∃!x A φ))
8 rmo5 2827 . 2 (∃*y A ψ ↔ (y A ψ∃!y A ψ))
96, 7, 83bitr4i 268 1 (∃*x A φ∃*y A ψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  Ⅎwnf 1544  ∃wrex 2615  ∃!wreu 2616  ∃*wrmo 2617 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622 This theorem is referenced by:  cbvrmov  2838
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