Theorem eqreu 3726

Description: A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)

Hypothesis

Ref	Expression
eqreu.1	⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
eqreu	⊢ ((𝐵 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜑)

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

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem eqreu

Step	Hyp	Ref	Expression
1		ralbiim 3114	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝑥 = 𝐵 → 𝜑)))
2		eqreu.1	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
3	2	ceqsralv 3514	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝑥 = 𝐵 → 𝜑) ↔ 𝜓))
4	3	anbi2d 630	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → ((∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝑥 = 𝐵 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵) ∧ 𝜓)))
5	1, 4	bitrid 283	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵) ∧ 𝜓)))
6		reu6i 3725	. . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜑)
7	6	ex 414	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵) → ∃!𝑥 ∈ 𝐴 𝜑))
8	5, 7	sylbird 260	. . 3 ⊢ (𝐵 ∈ 𝐴 → ((∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵) ∧ 𝜓) → ∃!𝑥 ∈ 𝐴 𝜑))
9	8	3impib 1117	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵) ∧ 𝜓) → ∃!𝑥 ∈ 𝐴 𝜑)
10	9	3com23 1127	1 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃!wreu 3375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-reu 3378

This theorem is referenced by:  uzwo3  12927  frmdup3  18748  frgpup3  19646  neiptopreu  22637  ufileu  23423  mirreu  27915  lmireu  28041  opreu2reuALT  31717  symgfcoeu  32243