Theorem bj-cleq 37017

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 6011	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶))
2		eleq2 2822	. . 3 ⊢ ((𝐴 “ 𝐶) = (𝐵 “ 𝐶) → ({𝑥} ∈ (𝐴 “ 𝐶) ↔ {𝑥} ∈ (𝐵 “ 𝐶)))
3	2	alrimiv 1928	. 2 ⊢ ((𝐴 “ 𝐶) = (𝐵 “ 𝐶) → ∀𝑥({𝑥} ∈ (𝐴 “ 𝐶) ↔ {𝑥} ∈ (𝐵 “ 𝐶)))
4		abbi 2798	. 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 206  ∀wal 1539   = wceq 1541   ∈ wcel 2113  {cab 2711  {csn 4577   “ cima 5624

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-ext 2705

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-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-cnv 5629  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634

This theorem is referenced by: (None)