Theorem sbcheg 38073

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 4085	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝐵 “ 𝐶) ⊆ 𝐶 ↔ ⦋𝐴 / 𝑥⦌(𝐵 “ 𝐶) ⊆ ⦋𝐴 / 𝑥⦌𝐶))
2		csbima12 5483	. . . . 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 3632	. . 3 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌(𝐵 “ 𝐶) ⊆ ⦋𝐴 / 𝑥⦌𝐶 ↔ (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌𝐶) ⊆ ⦋𝐴 / 𝑥⦌𝐶))
5	1, 4	bitrd 268	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝐵 “ 𝐶) ⊆ 𝐶 ↔ (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌𝐶) ⊆ ⦋𝐴 / 𝑥⦌𝐶))
6		df-he 38067	. . 3 ⊢ (𝐵 hereditary 𝐶 ↔ (𝐵 “ 𝐶) ⊆ 𝐶)
7	6	sbcbii 3491	. 2 ⊢ ([𝐴 / 𝑥]𝐵 hereditary 𝐶 ↔ [𝐴 / 𝑥](𝐵 “ 𝐶) ⊆ 𝐶)
8		df-he 38067	. 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵 hereditary ⦋𝐴 / 𝑥⦌𝐶 ↔ (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌𝐶) ⊆ ⦋𝐴 / 𝑥⦌𝐶)
9	5, 7, 8	3bitr4g 303	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 hereditary 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 hereditary ⦋𝐴 / 𝑥⦌𝐶))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ∈ wcel 1990  [wsbc 3435  ⦋csb 3533   ⊆ wss 3574   “ cima 5117   hereditary whe 38066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-he 38067

This theorem is referenced by:  frege77  38234