Theorem rmoi 3096

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 3095	. 2 ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → (𝐵 = 𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝜒)))
4	3	biimp3ar 1359	1 ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓) ∧ (𝐶 ∈ 𝐴 ∧ 𝜒)) → 𝐵 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 981   = wceq 1373   ∈ wcel 2177  ∃*wrmo 2488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rmo 2493  df-v 2775

This theorem is referenced by: (None)