Theorem sbcheg 44546

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 4487	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝐵 “ 𝐶) ⊆ 𝐶 ↔ ⦋𝐴 / 𝑥⦌(𝐵 “ 𝐶) ⊆ ⦋𝐴 / 𝑥⦌𝐶))
2		csbima12 6086	. . . . 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 3971	. . 3 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌(𝐵 “ 𝐶) ⊆ ⦋𝐴 / 𝑥⦌𝐶 ↔ (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌𝐶) ⊆ ⦋𝐴 / 𝑥⦌𝐶))
5	1, 4	bitrd 282	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝐵 “ 𝐶) ⊆ 𝐶 ↔ (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌𝐶) ⊆ ⦋𝐴 / 𝑥⦌𝐶))
6		df-he 44540	. . 3 ⊢ (𝐵 hereditary 𝐶 ↔ (𝐵 “ 𝐶) ⊆ 𝐶)
7	6	sbcbii 3803	. 2 ⊢ ([𝐴 / 𝑥]𝐵 hereditary 𝐶 ↔ [𝐴 / 𝑥](𝐵 “ 𝐶) ⊆ 𝐶)
8		df-he 44540	. 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵 hereditary ⦋𝐴 / 𝑥⦌𝐶 ↔ (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌𝐶) ⊆ ⦋𝐴 / 𝑥⦌𝐶)
9	5, 7, 8	3bitr4g 317	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 hereditary 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 hereditary ⦋𝐴 / 𝑥⦌𝐶))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2146  [wsbc 3747  ⦋csb 3856   ⊆ wss 3908   “ cima 5669   hereditary whe 44539

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5672  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-he 44540

This theorem is used by:  frege77  44707