Theorem bnj1316 34955

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 2886	. . . 4 ⊢ Ⅎ𝑥𝐴
3		bnj1316.2	. . . . 5 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵)
4	3	nfcii 2886	. . . 4 ⊢ Ⅎ𝑥𝐵
5	2, 4	nfeq 2911	. . 3 ⊢ Ⅎ𝑥 𝐴 = 𝐵
6	5	nf5ri 2201	. 2 ⊢ (𝐴 = 𝐵 → ∀𝑥 𝐴 = 𝐵)
7	6	bnj956 34911	1 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∪ ciun 4945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-rex 3060  df-iun 4947

This theorem is referenced by:  bnj1000  35076  bnj1318  35160