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Theorem rmob 3134
 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 2622 . 2 (∃*x A φ∃*x(x A φ))
2 simprl 732 . . . 4 ((∃*x(x A φ) (B A ψ)) → B A)
3 eleq1 2413 . . . 4 (B = C → (B AC A))
42, 3syl5ibcom 211 . . 3 ((∃*x(x A φ) (B A ψ)) → (B = CC A))
5 simpl 443 . . . 4 ((C A χ) → C A)
65a1i 10 . . 3 ((∃*x(x A φ) (B A ψ)) → ((C A χ) → C A))
7 simplrl 736 . . . . 5 (((∃*x(x A φ) (B A ψ)) C A) → B A)
8 simpr 447 . . . . 5 (((∃*x(x A φ) (B A ψ)) C A) → C A)
9 simpll 730 . . . . 5 (((∃*x(x A φ) (B A ψ)) C A) → ∃*x(x A φ))
10 simplrr 737 . . . . 5 (((∃*x(x A φ) (B A ψ)) C A) → ψ)
11 eleq1 2413 . . . . . . 7 (x = B → (x AB A))
12 rmoi.b . . . . . . 7 (x = B → (φψ))
1311, 12anbi12d 691 . . . . . 6 (x = B → ((x A φ) ↔ (B A ψ)))
14 eleq1 2413 . . . . . . 7 (x = C → (x AC A))
15 rmoi.c . . . . . . 7 (x = C → (φχ))
1614, 15anbi12d 691 . . . . . 6 (x = C → ((x A φ) ↔ (C A χ)))
1713, 16mob 3018 . . . . 5 (((B A C A) ∃*x(x A φ) (B A ψ)) → (B = C ↔ (C A χ)))
187, 8, 9, 7, 10, 17syl212anc 1192 . . . 4 (((∃*x(x A φ) (B A ψ)) C A) → (B = C ↔ (C A χ)))
1918ex 423 . . 3 ((∃*x(x A φ) (B A ψ)) → (C A → (B = C ↔ (C A χ))))
204, 6, 19pm5.21ndd 343 . 2 ((∃*x(x A φ) (B A ψ)) → (B = C ↔ (C A χ)))
211, 20sylanb 458 1 ((∃*x A φ (B A ψ)) → (B = C ↔ (C A χ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  ∃*wrmo 2617 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rmo 2622  df-v 2861 This theorem is referenced by:  rmoi  3135
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