Theorem bj-cleq 36309

Description: Substitution property for certain classes. (Contributed by BJ, 2-Apr-2019.)

Assertion

Ref	Expression
bj-cleq	⊢ (𝐴 = 𝐵 → {𝑥 ∣ {𝑥} ∈ (𝐴 “ 𝐶)} = {𝑥 ∣ {𝑥} ∈ (𝐵 “ 𝐶)})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem bj-cleq

Step	Hyp	Ref	Expression
1		imaeq1 6054	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶))
2		eleq2 2821	. . 3 ⊢ ((𝐴 “ 𝐶) = (𝐵 “ 𝐶) → ({𝑥} ∈ (𝐴 “ 𝐶) ↔ {𝑥} ∈ (𝐵 “ 𝐶)))
3	2	alrimiv 1929	. 2 ⊢ ((𝐴 “ 𝐶) = (𝐵 “ 𝐶) → ∀𝑥({𝑥} ∈ (𝐴 “ 𝐶) ↔ {𝑥} ∈ (𝐵 “ 𝐶)))
4		abbi 2799	. 2 ⊢ (∀𝑥({𝑥} ∈ (𝐴 “ 𝐶) ↔ {𝑥} ∈ (𝐵 “ 𝐶)) → {𝑥 ∣ {𝑥} ∈ (𝐴 “ 𝐶)} = {𝑥 ∣ {𝑥} ∈ (𝐵 “ 𝐶)})
5	1, 3, 4	3syl 18	1 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ {𝑥} ∈ (𝐴 “ 𝐶)} = {𝑥 ∣ {𝑥} ∈ (𝐵 “ 𝐶)})

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1538   = wceq 1540   ∈ wcel 2105  {cab 2708  {csn 4628   “ cima 5679

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689

This theorem is referenced by: (None)