Theorem bj-cleq 36334

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 6045	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶))
2		eleq2 2814	. . 3 ⊢ ((𝐴 “ 𝐶) = (𝐵 “ 𝐶) → ({𝑥} ∈ (𝐴 “ 𝐶) ↔ {𝑥} ∈ (𝐵 “ 𝐶)))
3	2	alrimiv 1922	. 2 ⊢ ((𝐴 “ 𝐶) = (𝐵 “ 𝐶) → ∀𝑥({𝑥} ∈ (𝐴 “ 𝐶) ↔ {𝑥} ∈ (𝐵 “ 𝐶)))
4		abbi 2792	. 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 1531   = wceq 1533   ∈ wcel 2098  {cab 2701  {csn 4621   “ cima 5670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680

This theorem is referenced by: (None)