Theorem sbeqalb 2969

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 239	. . . . 5 ⊢ ((𝜑 ↔ 𝑥 = 𝐴) → ((𝜑 ↔ 𝑥 = 𝐵) ↔ (𝑥 = 𝐴 ↔ 𝑥 = 𝐵)))
2	1	biimpa 294	. . . 4 ⊢ (((𝜑 ↔ 𝑥 = 𝐴) ∧ (𝜑 ↔ 𝑥 = 𝐵)) → (𝑥 = 𝐴 ↔ 𝑥 = 𝐵))
3	2	biimpd 143	. . 3 ⊢ (((𝜑 ↔ 𝑥 = 𝐴) ∧ (𝜑 ↔ 𝑥 = 𝐵)) → (𝑥 = 𝐴 → 𝑥 = 𝐵))
4	3	alanimi 1436	. 2 ⊢ ((∀𝑥(𝜑 ↔ 𝑥 = 𝐴) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝐵)) → ∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵))
5		sbceqal 2968	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵))
6	4, 5	syl5 32	1 ⊢ (𝐴 ∈ 𝑉 → ((∀𝑥(𝜑 ↔ 𝑥 = 𝐴) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝐵)) → 𝐴 = 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1330   = wceq 1332   ∈ wcel 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sbc 2914

This theorem is referenced by:  iotaval  5107