Theorem int-eqprincd 45012

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 7434	1 ⊢ (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐷))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  (class class class)co 7416   + caddc 11128

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-ov 7419

This theorem is used by: (None)