Theorem int-eqprincd 44428

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 7376	1 ⊢ (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐷))

Syntax hints:   → wi 4   = wceq 1541  (class class class)co 7358   + caddc 11029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-ov 7361

This theorem is referenced by: (None)