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Theorem cbvrmo 2501
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 2499 . . 3 (x A φy A ψ)
51, 2, 3cbvreu 2500 . . 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 2494 . 2 (∃*x A φ ↔ (x A φ∃!x A φ))
8 rmo5 2494 . 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 1322  wrex 2276  ∃!wreu 2277  ∃*wrmo 2278
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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-tru 1226  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-cleq 2006  df-clel 2009  df-nfc 2140  df-rex 2281  df-reu 2282  df-rmo 2283
This theorem is referenced by:  cbvrmov  2505  cbvdisj  3718
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