Theorem bj-cleq 38002

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 6052	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶))
2		eleq2 2852	. . 3 ⊢ ((𝐴 “ 𝐶) = (𝐵 “ 𝐶) → ({𝑥} ∈ (𝐴 “ 𝐶) ↔ {𝑥} ∈ (𝐵 “ 𝐶)))
3	2	alrimiv 1961	. 2 ⊢ ((𝐴 “ 𝐶) = (𝐵 “ 𝐶) → ∀𝑥({𝑥} ∈ (𝐴 “ 𝐶) ↔ {𝑥} ∈ (𝐵 “ 𝐶)))
4		abbi 2828	. 2 ⊢ (∀𝑥({𝑥} ∈ (𝐴 “ 𝐶) ↔ {𝑥} ∈ (𝐵 “ 𝐶)) → {𝑥 ∣ {𝑥} ∈ (𝐴 “ 𝐶)} = {𝑥 ∣ {𝑥} ∈ (𝐵 “ 𝐶)})
5	1, 3, 4	3syl 19	1 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ {𝑥} ∈ (𝐴 “ 𝐶)} = {𝑥 ∣ {𝑥} ∈ (𝐵 “ 𝐶)})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567   = wceq 1569   ∈ wcel 2146  {cab 2741  {csn 4587   “ cima 5657

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667

This theorem is used by: (None)