Theorem bj-elssuniab 7037

Description: Version of elssuni 3582 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)

Hypothesis

Ref	Expression
bj-elssuniab.nf	⊢ ℲxA

Assertion

Ref	Expression
bj-elssuniab	⊢ (A ∈ 𝑉 → ([A / x]φ → A ⊆ ∪ {x ∣ φ}))

Proof of Theorem bj-elssuniab

Step	Hyp	Ref	Expression
1		sbc8g 2748	. 2 ⊢ (A ∈ 𝑉 → ([A / x]φ ↔ A ∈ {x ∣ φ}))
2		elssuni 3582	. 2 ⊢ (A ∈ {x ∣ φ} → A ⊆ ∪ {x ∣ φ})
3	1, 2	syl6bi 152	1 ⊢ (A ∈ 𝑉 → ([A / x]φ → A ⊆ ∪ {x ∣ φ}))

Syntax hints:   → wi 4   ∈ wcel 1374  {cab 2008  Ⅎwnfc 2147  [wsbc 2741   ⊆ wss 2894  ∪ cuni 3554

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-sbc 2742  df-in 2901  df-ss 2908  df-uni 3555

This theorem is referenced by: (None)