Theorem bnj1316 35520

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1316.1	⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
bnj1316.2	⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵)

Assertion

Ref	Expression
bnj1316	⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)

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

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥, 𝑦)

Proof of Theorem bnj1316

Step	Hyp	Ref	Expression
1		bnj1316.1	. . . . 5 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
2	1	nfcii 2915	. . . 4 ⊢ Ⅎ𝑥𝐴
3		bnj1316.2	. . . . 5 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵)
4	3	nfcii 2915	. . . 4 ⊢ Ⅎ𝑥𝐵
5	2, 4	nfeq 2939	. . 3 ⊢ Ⅎ𝑥 𝐴 = 𝐵
6	5	nf5ri 2235	. 2 ⊢ (𝐴 = 𝐵 → ∀𝑥 𝐴 = 𝐵)
7	6	bnj956 35477	1 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568   = wceq 1570   ∈ wcel 2147  ∪ ciun 4954

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-rex 3091  df-iun 4956

This theorem is used by:  bnj1000  35641  bnj1318  35725