Theorem int-eqprincd 44160

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 7371	1 ⊢ (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐷))

Syntax hints:   → wi 4   = wceq 1540  (class class class)co 7353   + caddc 11031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-iota 6442  df-fv 6494  df-ov 7356

This theorem is referenced by: (None)