Theorem bj-elssuniab 16492

Description: Version of elssuni 3926 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 3040	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}))
2		elssuni 3926	. 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 2202  {cab 2217  Ⅎwnfc 2362  [wsbc 3032   ⊆ wss 3201  ∪ cuni 3898

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-sbc 3033  df-in 3207  df-ss 3214  df-uni 3899

This theorem is referenced by: (None)