Theorem sbeqalb 3844

Description: Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.)

Assertion

Ref	Expression
sbeqalb	⊢ (𝐴 ∈ 𝑉 → ((∀𝑥(𝜑 ↔ 𝑥 = 𝐴) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝐵)) → 𝐴 = 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

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

Proof of Theorem sbeqalb

Step	Hyp	Ref	Expression
1		bibi1 350	. . . . 5 ⊢ ((𝜑 ↔ 𝑥 = 𝐴) → ((𝜑 ↔ 𝑥 = 𝐵) ↔ (𝑥 = 𝐴 ↔ 𝑥 = 𝐵)))
2	1	biimpa 475	. . . 4 ⊢ (((𝜑 ↔ 𝑥 = 𝐴) ∧ (𝜑 ↔ 𝑥 = 𝐵)) → (𝑥 = 𝐴 ↔ 𝑥 = 𝐵))
3	2	biimpd 228	. . 3 ⊢ (((𝜑 ↔ 𝑥 = 𝐴) ∧ (𝜑 ↔ 𝑥 = 𝐵)) → (𝑥 = 𝐴 → 𝑥 = 𝐵))
4	3	alanimi 1811	. 2 ⊢ ((∀𝑥(𝜑 ↔ 𝑥 = 𝐴) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝐵)) → ∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵))
5		sbceqal 3842	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵))
6	4, 5	syl5 34	1 ⊢ (𝐴 ∈ 𝑉 → ((∀𝑥(𝜑 ↔ 𝑥 = 𝐴) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝐵)) → 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394  ∀wal 1532   = wceq 1534   ∈ wcel 2099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464

This theorem is referenced by:  iotavalOLD  6528