Theorem csbfvg 5534

Description: Substitution for a function value. (Contributed by NM, 1-Jan-2006.)

Assertion

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

Distinct variable group:   𝑥,𝐹

Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem csbfvg

Step	Hyp	Ref	Expression
1		csbfv2g 5533	. 2 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘⦋𝐴 / 𝑥⦌𝑥))
2		csbvarg 3077	. . 3 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴)
3	2	fveq2d 5500	. 2 ⊢ (𝐴 ∈ 𝐶 → (𝐹‘⦋𝐴 / 𝑥⦌𝑥) = (𝐹‘𝐴))
4	1, 3	eqtrd 2203	1 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘𝐴))

Syntax hints:   → wi 4   = wceq 1348   ∈ wcel 2141  ⦋csb 3049  ‘cfv 5198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206

This theorem is referenced by:  ixpsnval  6679