Theorem bj-intabssel 15281

Description: Version of intss1 3885 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)

Hypothesis

Ref	Expression
bj-intabssel.nf	⊢ Ⅎ𝑥𝐴

Assertion

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

Proof of Theorem bj-intabssel

Step	Hyp	Ref	Expression
1		bj-intabssel.nf	. . 3 ⊢ Ⅎ𝑥𝐴
2	1	nfsbc1 3003	. . 3 ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑
3		sbceq1a 2995	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
4	1, 2, 3	elabgf 2902	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ [𝐴 / 𝑥]𝜑))
5		intss1 3885	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴)
6	4, 5	biimtrrdi 164	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2164  {cab 2179  Ⅎwnfc 2323  [wsbc 2985   ⊆ wss 3153  ∩ cint 3870

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-sbc 2986  df-in 3159  df-ss 3166  df-int 3871

This theorem is referenced by: (None)