Theorem parteq1d 40079

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 40075	. 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 39422

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-ext 2735  ax-sep 5255  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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ec 8757  df-qs 8761  df-coss 39699  df-cnvrefrel 39805  df-dmqs 39921  df-funALTV 39965  df-disjALTV 39988  df-part 40067

This theorem is used by: (None)