Theorem rmoi 3853

Description: Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)

Hypotheses

Ref	Expression
rmoi.b	⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
rmoi.c	⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜒))

Assertion

Ref	Expression
rmoi	⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓) ∧ (𝐶 ∈ 𝐴 ∧ 𝜒)) → 𝐵 = 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜓,𝑥   𝜒,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rmoi

Step	Hyp	Ref	Expression
1		rmoi.b	. . 3 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
2		rmoi.c	. . 3 ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜒))
3	1, 2	rmob 3852	. 2 ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → (𝐵 = 𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝜒)))
4	3	biimp3ar 1496	1 ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓) ∧ (𝐶 ∈ 𝐴 ∧ 𝜒)) → 𝐵 = 𝐶)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∃*wrmo 3375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-clab 2748  df-cleq 2761  df-clel 2844  df-rmo 3376  df-v 3465

This theorem is referenced by:  eqsqrtd  15415  efgred2  19819  0frgp  19845  frgpnabllem2  19940  frgpcyg  21688  cdleme0moN  40884  proot1mul  43808