Theorem int-eqprincd 41751

Description: PrincipleOfEquality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)

Hypotheses

Ref	Expression
int-eqprincd.1	⊢ (𝜑 → 𝐴 = 𝐵)
int-eqprincd.2	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
int-eqprincd	⊢ (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐷))

Proof of Theorem int-eqprincd

Step	Hyp	Ref	Expression
1		int-eqprincd.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		int-eqprincd.2	. 2 ⊢ (𝜑 → 𝐶 = 𝐷)
3	1, 2	oveq12d 7286	1 ⊢ (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐷))

Syntax hints:   → wi 4   = wceq 1541  (class class class)co 7268   + caddc 10858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-iota 6388  df-fv 6438  df-ov 7271

This theorem is referenced by: (None)