Theorem moi 2943

Description: Equality implied by "at most one". (Contributed by NM, 18-Feb-2006.)

Hypotheses

Ref	Expression
moi.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
moi.2	⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))

Assertion

Ref	Expression
moi	⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∃*𝑥𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝐴 = 𝐵)

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

Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem moi

Step	Hyp	Ref	Expression
1		moi.1	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2		moi.2	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))
3	1, 2	mob 2942	. . . . 5 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∃*𝑥𝜑 ∧ 𝜓) → (𝐴 = 𝐵 ↔ 𝜒))
4	3	biimprd 158	. . . 4 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∃*𝑥𝜑 ∧ 𝜓) → (𝜒 → 𝐴 = 𝐵))
5	4	3expia 1207	. . 3 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∃*𝑥𝜑) → (𝜓 → (𝜒 → 𝐴 = 𝐵)))
6	5	impd 254	. 2 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∃*𝑥𝜑) → ((𝜓 ∧ 𝜒) → 𝐴 = 𝐵))
7	6	3impia 1202	1 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∃*𝑥𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364  ∃*wmo 2043   ∈ wcel 2164

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762

This theorem is referenced by: (None)