Theorem csbfv12g 5679

Description: Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.)

Assertion

Ref	Expression
csbfv12g	⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵))

Proof of Theorem csbfv12g

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbiotag 5319	. . 3 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(℩𝑦𝐵𝐹𝑦) = (℩𝑦[𝐴 / 𝑥]𝐵𝐹𝑦))
2		sbcbrg 4143	. . . . 5 ⊢ (𝐴 ∈ 𝐶 → ([𝐴 / 𝑥]𝐵𝐹𝑦 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦))
3		csbconstg 3141	. . . . . 6 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦)
4	3	breq2d 4100	. . . . 5 ⊢ (𝐴 ∈ 𝐶 → (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹𝑦))
5	2, 4	bitrd 188	. . . 4 ⊢ (𝐴 ∈ 𝐶 → ([𝐴 / 𝑥]𝐵𝐹𝑦 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹𝑦))
6	5	iotabidv 5309	. . 3 ⊢ (𝐴 ∈ 𝐶 → (℩𝑦[𝐴 / 𝑥]𝐵𝐹𝑦) = (℩𝑦⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹𝑦))
7	1, 6	eqtrd 2264	. 2 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(℩𝑦𝐵𝐹𝑦) = (℩𝑦⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹𝑦))
8		df-fv 5334	. . 3 ⊢ (𝐹‘𝐵) = (℩𝑦𝐵𝐹𝑦)
9	8	csbeq2i 3154	. 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ⦋𝐴 / 𝑥⦌(℩𝑦𝐵𝐹𝑦)
10		df-fv 5334	. 2 ⊢ (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) = (℩𝑦⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹𝑦)
11	7, 9, 10	3eqtr4g 2289	1 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵))

Syntax hints:   → wi 4   = wceq 1397   ∈ wcel 2202  [wsbc 3031  ⦋csb 3127   class class class wbr 4088  ℩cio 5284  ‘cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334

This theorem is referenced by:  csbfv2g  5680