Theorem bj-elssuniab 16113

Description: Version of elssuni 3915 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 3036	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}))
2		elssuni 3915	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝐴 ⊆ ∪ {𝑥 ∣ 𝜑})
3	1, 2	biimtrdi 163	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 → 𝐴 ⊆ ∪ {𝑥 ∣ 𝜑}))

Syntax hints:   → wi 4   ∈ wcel 2200  {cab 2215  Ⅎwnfc 2359  [wsbc 3028   ⊆ wss 3197  ∪ cuni 3887

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sbc 3029  df-in 3203  df-ss 3210  df-uni 3888

This theorem is referenced by: (None)