Theorem eqreu 3690

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 3128	. . . . 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 3494	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝑥 = 𝐵 → 𝜑) ↔ 𝜓))
4	3	anbi2d 642	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → ((∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝑥 = 𝐵 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵) ∧ 𝜓)))
5	1, 4	bitrid 286	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵) ∧ 𝜓)))
6		reu6i 3689	. . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜑)
7	6	ex 418	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵) → ∃!𝑥 ∈ 𝐴 𝜑))
8	5, 7	sylbird 263	. . 3 ⊢ (𝐵 ∈ 𝐴 → ((∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵) ∧ 𝜓) → ∃!𝑥 ∈ 𝐴 𝜑))
9	8	3impib 1134	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵) ∧ 𝜓) → ∃!𝑥 ∈ 𝐴 𝜑)
10	9	3com23 1144	1 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃!wreu 3367

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-reu 3370

This theorem is used by:  uzwo3  13110  frmdup3  19113  frgpup3  20043  neiptopreu  23506  ufileu  24293  mirreu  29193  lmireu  29352  opreu2reuALT  33172  ccatws1f1o  33610  symgfcoeu  33742  aks6d1c7lem4  43499