Theorem sbeqalb 3019

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 240	. . . . 5 ⊢ ((𝜑 ↔ 𝑥 = 𝐴) → ((𝜑 ↔ 𝑥 = 𝐵) ↔ (𝑥 = 𝐴 ↔ 𝑥 = 𝐵)))
2	1	biimpa 296	. . . 4 ⊢ (((𝜑 ↔ 𝑥 = 𝐴) ∧ (𝜑 ↔ 𝑥 = 𝐵)) → (𝑥 = 𝐴 ↔ 𝑥 = 𝐵))
3	2	biimpd 144	. . 3 ⊢ (((𝜑 ↔ 𝑥 = 𝐴) ∧ (𝜑 ↔ 𝑥 = 𝐵)) → (𝑥 = 𝐴 → 𝑥 = 𝐵))
4	3	alanimi 1459	. 2 ⊢ ((∀𝑥(𝜑 ↔ 𝑥 = 𝐴) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝐵)) → ∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵))
5		sbceqal 3018	. 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 104   ↔ wb 105  ∀wal 1351   = wceq 1353   ∈ wcel 2148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sbc 2963

This theorem is referenced by:  iotaval  5188