Theorem sbcheg 40118

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 4463	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝐵 “ 𝐶) ⊆ 𝐶 ↔ ⦋𝐴 / 𝑥⦌(𝐵 “ 𝐶) ⊆ ⦋𝐴 / 𝑥⦌𝐶))
2		csbima12 5942	. . . . 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 3998	. . 3 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌(𝐵 “ 𝐶) ⊆ ⦋𝐴 / 𝑥⦌𝐶 ↔ (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌𝐶) ⊆ ⦋𝐴 / 𝑥⦌𝐶))
5	1, 4	bitrd 281	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝐵 “ 𝐶) ⊆ 𝐶 ↔ (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌𝐶) ⊆ ⦋𝐴 / 𝑥⦌𝐶))
6		df-he 40112	. . 3 ⊢ (𝐵 hereditary 𝐶 ↔ (𝐵 “ 𝐶) ⊆ 𝐶)
7	6	sbcbii 3829	. 2 ⊢ ([𝐴 / 𝑥]𝐵 hereditary 𝐶 ↔ [𝐴 / 𝑥](𝐵 “ 𝐶) ⊆ 𝐶)
8		df-he 40112	. 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵 hereditary ⦋𝐴 / 𝑥⦌𝐶 ↔ (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌𝐶) ⊆ ⦋𝐴 / 𝑥⦌𝐶)
9	5, 7, 8	3bitr4g 316	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 hereditary 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 hereditary ⦋𝐴 / 𝑥⦌𝐶))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1533   ∈ wcel 2110  [wsbc 3772  ⦋csb 3883   ⊆ wss 3936   “ cima 5553   hereditary whe 40111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060  df-opab 5122  df-xp 5556  df-cnv 5558  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-he 40112

This theorem is referenced by:  frege77  40279