Theorem rmoi 3135

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 3134	. 2 ⊢ ((∃*x ∈ A φ ∧ (B ∈ A ∧ ψ)) → (B = C ↔ (C ∈ A ∧ χ)))
4	3	biimp3ar 1282	1 ⊢ ((∃*x ∈ A φ ∧ (B ∈ A ∧ ψ) ∧ (C ∈ A ∧ χ)) → B = C)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∃*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-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: (None)