Theorem moi2 3657

Description: Consequence of "at most one". (Contributed by NM, 29-Jun-2008.)

Hypothesis

Ref	Expression
moi2.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
moi2	⊢ (((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑) ∧ (𝜑 ∧ 𝜓)) → 𝑥 = 𝐴)

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem moi2

Step	Hyp	Ref	Expression
1		moi2.1	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2	1	mob2 3656	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑 ∧ 𝜑) → (𝑥 = 𝐴 ↔ 𝜓))
3	2	3expa 1124	. . 3 ⊢ (((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑) ∧ 𝜑) → (𝑥 = 𝐴 ↔ 𝜓))
4	3	biimprd 249	. 2 ⊢ (((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑) ∧ 𝜑) → (𝜓 → 𝑥 = 𝐴))
5	4	impr 455	1 ⊢ (((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑) ∧ (𝜑 ∧ 𝜓)) → 𝑥 = 𝐴)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃*wmo 2541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433

This theorem is referenced by:  fsum  15673  fprod  15897  txcn  23609  haustsms2  24120