Theorem moi 3009

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 3008	. . . . 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 1236	. . 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 1231	1 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∃*𝑥𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402  ∃*wmo 2087   ∈ wcel 2209

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823

This theorem is referenced by: (None)