Theorem bj-elssuniab 13672

Description: Version of elssuni 3817 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 2958	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}))
2		elssuni 3817	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝐴 ⊆ ∪ {𝑥 ∣ 𝜑})
3	1, 2	syl6bi 162	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 → 𝐴 ⊆ ∪ {𝑥 ∣ 𝜑}))

Syntax hints:   → wi 4   ∈ wcel 2136  {cab 2151  Ⅎwnfc 2295  [wsbc 2951   ⊆ wss 3116  ∪ cuni 3789

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952  df-in 3122  df-ss 3129  df-uni 3790

This theorem is referenced by: (None)