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Mirrors  >  Home  >  ILE Home  >  Th. List  >  rmob Structured version   GIF version

Theorem rmob 2844
Description: Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.)
Hypotheses
Ref Expression
rmoi.b (x = B → (φψ))
rmoi.c (x = 𝐶 → (φχ))
Assertion
Ref Expression
rmob ((∃*x A φ (B A ψ)) → (B = 𝐶 ↔ (𝐶 A χ)))
Distinct variable groups:   x,A   x,B   x,𝐶   ψ,x   χ,x
Allowed substitution hint:   φ(x)

Proof of Theorem rmob
StepHypRef Expression
1 df-rmo 2308 . 2 (∃*x A φ∃*x(x A φ))
2 simprl 483 . . . 4 ((∃*x(x A φ) (B A ψ)) → B A)
3 eleq1 2097 . . . 4 (B = 𝐶 → (B A𝐶 A))
42, 3syl5ibcom 144 . . 3 ((∃*x(x A φ) (B A ψ)) → (B = 𝐶𝐶 A))
5 simpl 102 . . . 4 ((𝐶 A χ) → 𝐶 A)
65a1i 9 . . 3 ((∃*x(x A φ) (B A ψ)) → ((𝐶 A χ) → 𝐶 A))
7 simplrl 487 . . . . 5 (((∃*x(x A φ) (B A ψ)) 𝐶 A) → B A)
8 simpr 103 . . . . 5 (((∃*x(x A φ) (B A ψ)) 𝐶 A) → 𝐶 A)
9 simpll 481 . . . . 5 (((∃*x(x A φ) (B A ψ)) 𝐶 A) → ∃*x(x A φ))
10 simplrr 488 . . . . 5 (((∃*x(x A φ) (B A ψ)) 𝐶 A) → ψ)
11 eleq1 2097 . . . . . . 7 (x = B → (x AB A))
12 rmoi.b . . . . . . 7 (x = B → (φψ))
1311, 12anbi12d 442 . . . . . 6 (x = B → ((x A φ) ↔ (B A ψ)))
14 eleq1 2097 . . . . . . 7 (x = 𝐶 → (x A𝐶 A))
15 rmoi.c . . . . . . 7 (x = 𝐶 → (φχ))
1614, 15anbi12d 442 . . . . . 6 (x = 𝐶 → ((x A φ) ↔ (𝐶 A χ)))
1713, 16mob 2717 . . . . 5 (((B A 𝐶 A) ∃*x(x A φ) (B A ψ)) → (B = 𝐶 ↔ (𝐶 A χ)))
187, 8, 9, 7, 10, 17syl212anc 1144 . . . 4 (((∃*x(x A φ) (B A ψ)) 𝐶 A) → (B = 𝐶 ↔ (𝐶 A χ)))
1918ex 108 . . 3 ((∃*x(x A φ) (B A ψ)) → (𝐶 A → (B = 𝐶 ↔ (𝐶 A χ))))
204, 6, 19pm5.21ndd 620 . 2 ((∃*x(x A φ) (B A ψ)) → (B = 𝐶 ↔ (𝐶 A χ)))
211, 20sylanb 268 1 ((∃*x A φ (B A ψ)) → (B = 𝐶 ↔ (𝐶 A χ)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  ∃*wmo 1898  ∃*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-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rmo 2308  df-v 2553
This theorem is referenced by:  rmoi  2845
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