Theorem rmoi 2845

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 2844	. 2 ⊢ ((∃*x ∈ A φ ∧ (B ∈ A ∧ ψ)) → (B = 𝐶 ↔ (𝐶 ∈ A ∧ χ)))
4	3	biimp3ar 1235	1 ⊢ ((∃*x ∈ A φ ∧ (B ∈ A ∧ ψ) ∧ (𝐶 ∈ A ∧ χ)) → B = 𝐶)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  ∃*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: (None)