Theorem qseq12d 42820

Description: Equality theorem for quotient set, deduction form. (Contributed by Steven Nguyen, 30-Apr-2023.)

Hypotheses

Ref	Expression
qseq12d.1	⊢ (𝜑 → 𝐴 = 𝐵)
qseq12d.2	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
qseq12d	⊢ (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐷))

Proof of Theorem qseq12d

Step	Hyp	Ref	Expression
1		qseq12d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		qseq12d.2	. 2 ⊢ (𝜑 → 𝐶 = 𝐷)
3		qseq12 8738	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 / 𝐶) = (𝐵 / 𝐷))
4	1, 2, 3	syl2anc 593	1 ⊢ (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐷))

Syntax hints:   → wi 4   = wceq 1559   / cqs 8672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-cnv 5653  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-ec 8675  df-qs 8679

This theorem is referenced by:  prjspval  43149