Theorem rmoi 3126

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 3125	. 2 ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → (𝐵 = 𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝜒)))
4	3	biimp3ar 1382	1 ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓) ∧ (𝐶 ∈ 𝐴 ∧ 𝜒)) → 𝐵 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  ∃*wrmo 2513

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rmo 2518  df-v 2804

This theorem is referenced by: (None)