Theorem eldisjeqd 38851

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

Hypothesis

Ref	Expression
eldisjeqd.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
eldisjeqd	⊢ (𝜑 → ( ElDisj 𝐴 ↔ ElDisj 𝐵))

Proof of Theorem eldisjeqd

Step	Hyp	Ref	Expression
1		eldisjeqd.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		eldisjeq 38849	. 2 ⊢ (𝐴 = 𝐵 → ( ElDisj 𝐴 ↔ ElDisj 𝐵))
3	1, 2	syl 17	1 ⊢ (𝜑 → ( ElDisj 𝐴 ↔ ElDisj 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ElDisj weldisj 38268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-coss 38523  df-cnvrefrel 38629  df-funALTV 38790  df-disjALTV 38813  df-eldisj 38815

This theorem is referenced by: (None)