Theorem eldisjeqi 39216

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 39215	. 2 ⊢ (𝐴 = 𝐵 → ( ElDisj 𝐴 ↔ ElDisj 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ( ElDisj 𝐴 ↔ ElDisj 𝐵)

Syntax hints:   ↔ wb 207   = wceq 1547   ElDisj weldisj 38595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-11 2168  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-coss 38875  df-cnvrefrel 38981  df-funALTV 39141  df-disjALTV 39164  df-eldisj 39166

This theorem is referenced by:  eldisjs7  39315