Theorem parteq1d 38776

Description: Equality theorem for partition, deduction version. (Contributed by Peter Mazsa, 5-Oct-2021.)

Hypothesis

Ref	Expression
parteq1d.1	⊢ (𝜑 → 𝑅 = 𝑆)

Assertion

Ref	Expression
parteq1d	⊢ (𝜑 → (𝑅 Part 𝐴 ↔ 𝑆 Part 𝐴))

Proof of Theorem parteq1d

Step	Hyp	Ref	Expression
1		parteq1d.1	. 2 ⊢ (𝜑 → 𝑅 = 𝑆)
2		parteq1 38772	. 2 ⊢ (𝑅 = 𝑆 → (𝑅 Part 𝐴 ↔ 𝑆 Part 𝐴))
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝑅 Part 𝐴 ↔ 𝑆 Part 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   Part wpart 38214

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-11 2158  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

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-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ec 8627  df-qs 8631  df-coss 38408  df-cnvrefrel 38524  df-dmqs 38636  df-funALTV 38680  df-disjALTV 38703  df-part 38764

This theorem is referenced by: (None)