Theorem itgeq1d 45878

Description: Equality theorem for an integral. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypothesis

Ref	Expression
itgeq1d.aeqb	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
itgeq1d	⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

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

Proof of Theorem itgeq1d

Step	Hyp	Ref	Expression
1		itgeq1d.aeqb	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		itgeq1 25828	. 2 ⊢ (𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)
3	1, 2	syl 17	1 ⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)

Syntax hints:   → wi 4   = wceq 1537  ∫citg 25672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-xp 5706  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-iota 6525  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-seq 14053  df-sum 15735  df-itg 25677

This theorem is referenced by:  itgspltprt  45900  fourierdlem73  46100  fourierdlem81  46108  fourierdlem92  46119  fourierdlem93  46120  fourierdlem103  46130  fourierdlem104  46131  fourierdlem107  46134  fourierdlem109  46136  fourierdlem111  46138