Theorem disjeqd 39168

Description: Equality theorem for disjoints, deduction version. (Contributed by Peter Mazsa, 22-Sep-2021.)

Hypothesis

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

Assertion

Ref	Expression
disjeqd	⊢ (𝜑 → ( Disj 𝐴 ↔ Disj 𝐵))

Proof of Theorem disjeqd

Step	Hyp	Ref	Expression
1		disjeqd.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		disjeq 39166	. 2 ⊢ (𝐴 = 𝐵 → ( Disj 𝐴 ↔ Disj 𝐵))
3	1, 2	syl 17	1 ⊢ (𝜑 → ( Disj 𝐴 ↔ Disj 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   Disj wdisjALTV 38551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709  ax-sep 5231  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-coss 38833  df-cnvrefrel 38939  df-funALTV 39099  df-disjALTV 39122

This theorem is referenced by:  eldisjeq  39173  eqvrelqseqdisj3  39277