Theorem parteq1d 39629

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 39625	. 2 ⊢ (𝑅 = 𝑆 → (𝑅 Part 𝐴 ↔ 𝑆 Part 𝐴))
3	1, 2	syl 18	1 ⊢ (𝜑 → (𝑅 Part 𝐴 ↔ 𝑆 Part 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   Part wpart 38972

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-11 2194  ax-ext 2732  ax-sep 5251  ax-pr 5398

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 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ec 8698  df-qs 8702  df-coss 39249  df-cnvrefrel 39355  df-dmqs 39471  df-funALTV 39515  df-disjALTV 39538  df-part 39617

This theorem is used by: (None)