Theorem eqreu 3365

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 3051	. . . . 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 3207	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝑥 = 𝐵 → 𝜑) ↔ 𝜓))
4	3	anbi2d 736	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → ((∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝑥 = 𝐵 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵) ∧ 𝜓)))
5	1, 4	syl5bb 271	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵) ∧ 𝜓)))
6		reu6i 3364	. . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜑)
7	6	ex 449	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵) → ∃!𝑥 ∈ 𝐴 𝜑))
8	5, 7	sylbird 249	. . 3 ⊢ (𝐵 ∈ 𝐴 → ((∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵) ∧ 𝜓) → ∃!𝑥 ∈ 𝐴 𝜑))
9	8	3impib 1254	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵) ∧ 𝜓) → ∃!𝑥 ∈ 𝐴 𝜑)
10	9	3com23 1263	1 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃!wreu 2898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-reu 2903  df-v 3175

This theorem is referenced by:  uzwo3  11659  frmdup3  17227  frgpup3  18014  neiptopreu  20747  ufileu  21533  mirreu  25359  lmireu  25482  symgfcoeu  29176