Theorem eldisjeqi 36780

Description: Equality theorem for disjoint elementhood, inference version. (Contributed by Peter Mazsa, 23-Sep-2021.)

Hypothesis

Ref	Expression
eldisjeqi.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
eldisjeqi	⊢ ( ElDisj 𝐴 ↔ ElDisj 𝐵)

Proof of Theorem eldisjeqi

Step	Hyp	Ref	Expression
1		eldisjeqi.1	. 2 ⊢ 𝐴 = 𝐵
2		eldisjeq 36779	. 2 ⊢ (𝐴 = 𝐵 → ( ElDisj 𝐴 ↔ ElDisj 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ( ElDisj 𝐴 ↔ ElDisj 𝐵)

Syntax hints:   ↔ wb 205   = wceq 1539   ElDisj weldisj 36296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-coss 36464  df-cnvrefrel 36570  df-funALTV 36720  df-disjALTV 36743  df-eldisj 36745

This theorem is referenced by: (None)