Theorem sbceqal 3001

Description: A variation of extensionality for classes. (Contributed by Andrew Salmon, 28-Jun-2011.)

Assertion

Ref	Expression
sbceqal	⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem sbceqal

Step	Hyp	Ref	Expression
1		spsbc 2957	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → [𝐴 / 𝑥](𝑥 = 𝐴 → 𝑥 = 𝐵)))
2		sbcimg 2987	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑥 = 𝐴 → 𝑥 = 𝐵) ↔ ([𝐴 / 𝑥]𝑥 = 𝐴 → [𝐴 / 𝑥]𝑥 = 𝐵)))
3		eqid 2164	. . . . 5 ⊢ 𝐴 = 𝐴
4		eqsbc3 2985	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐴 ↔ 𝐴 = 𝐴))
5	3, 4	mpbiri 167	. . . 4 ⊢ (𝐴 ∈ 𝑉 → [𝐴 / 𝑥]𝑥 = 𝐴)
6		pm5.5 241	. . . 4 ⊢ ([𝐴 / 𝑥]𝑥 = 𝐴 → (([𝐴 / 𝑥]𝑥 = 𝐴 → [𝐴 / 𝑥]𝑥 = 𝐵) ↔ [𝐴 / 𝑥]𝑥 = 𝐵))
7	5, 6	syl 14	. . 3 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝑥 = 𝐴 → [𝐴 / 𝑥]𝑥 = 𝐵) ↔ [𝐴 / 𝑥]𝑥 = 𝐵))
8		eqsbc3 2985	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
9	2, 7, 8	3bitrd 213	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑥 = 𝐴 → 𝑥 = 𝐵) ↔ 𝐴 = 𝐵))
10	1, 9	sylibd 148	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1340   = wceq 1342   ∈ wcel 2135  [wsbc 2946

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-sbc 2947

This theorem is referenced by:  sbeqalb  3002  snsssn  3735