Theorem sbcheg 43768

Description: Distribute proper substitution through herditary relation. (Contributed by RP, 29-Jun-2020.)

Assertion

Ref	Expression
sbcheg	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 hereditary 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 hereditary ⦋𝐴 / 𝑥⦌𝐶))

Proof of Theorem sbcheg

Step	Hyp	Ref	Expression
1		sbcssg 4483	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝐵 “ 𝐶) ⊆ 𝐶 ↔ ⦋𝐴 / 𝑥⦌(𝐵 “ 𝐶) ⊆ ⦋𝐴 / 𝑥⦌𝐶))
2		csbima12 6050	. . . . 5 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 “ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌𝐶)
3	2	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 “ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌𝐶))
4	3	sseq1d 3978	. . 3 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌(𝐵 “ 𝐶) ⊆ ⦋𝐴 / 𝑥⦌𝐶 ↔ (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌𝐶) ⊆ ⦋𝐴 / 𝑥⦌𝐶))
5	1, 4	bitrd 279	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝐵 “ 𝐶) ⊆ 𝐶 ↔ (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌𝐶) ⊆ ⦋𝐴 / 𝑥⦌𝐶))
6		df-he 43762	. . 3 ⊢ (𝐵 hereditary 𝐶 ↔ (𝐵 “ 𝐶) ⊆ 𝐶)
7	6	sbcbii 3810	. 2 ⊢ ([𝐴 / 𝑥]𝐵 hereditary 𝐶 ↔ [𝐴 / 𝑥](𝐵 “ 𝐶) ⊆ 𝐶)
8		df-he 43762	. 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵 hereditary ⦋𝐴 / 𝑥⦌𝐶 ↔ (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌𝐶) ⊆ ⦋𝐴 / 𝑥⦌𝐶)
9	5, 7, 8	3bitr4g 314	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 hereditary 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 hereditary ⦋𝐴 / 𝑥⦌𝐶))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  [wsbc 3753  ⦋csb 3862   ⊆ wss 3914   “ cima 5641   hereditary whe 43761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-he 43762

This theorem is referenced by:  frege77  43929